Formal & Informal Geometry
Formal & Informal Geometry MATH 3305
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Date Created: 09/19/15
Euclidean Topics Sets Triangles Set of all Triangles Subsets focused on sides Subsets focused on angles Areas and altitudes of similar triangles Recognizing similarity Inequality Theorems Pythagorean Theorem Convex Quadrilaterals Arbitrary Quadrilaterals Trapezoids Parallelo grams Rhombus Rectangle Square Circles Inscribed angle theorem Tangent lines to a circle Tangent circles Tangent lines to a pair of circles Answers to Exercises Appendix A Theorems Appendix B SMSG Axioms for Euclidean Geometry Appendix C Glossary Sets A set is a mathematical object 7 a collection of objects called set elements that have been de ned as having a shared property It is a requirement that a reasonable observer be able to tell whether or not any object belongs to the set at had 7 this is a property called well defined For example the set of all capital letters in the English alphabet is welldefined A B Z are set elements while 1 E and my puppy Elle do not belong to it Any reasonable person can tell what is or is not in this set We can name this set A Sets have subsets 7 another set that is composed of some group of set elements from the original set It is not a requirement that the subset be distinguishable from the original set A set is a subset of itself But if the original set has at least one more element than the subset we note this by saying the that subset is a proper subset For example the set of capital vowels A E I O U is a proper subset of the set A One thing to notice about subsets They inherit properties from the original set If a set has a property then the subset has that property too For example P prime numbers is a well de ned set B 3 5 7 11 is a proper subset of P And all the elements of B inherited the property that they re prime from P If you have 2 sets it may be that they share an element or two This means that the intersection 7 or common part 7 of the two sets has something in it For example A l a 53 and B l b 5 JH A and B share the elements 1 These two elements are the intersection of A and B It is possible that there are no shared elements Given C z 35 7 Note that A and C share no elements we say A and C are disjoint sets You may arrange the elements of a set in any way you like to work with the set One common method of arranging the set elements is to partition the set into disjoint subsets This is a useful way to handle information With a partition each set element belongs to exactly one subset each subset is disjoint from all the others and taken together all the partition subsets merge back to being the original set For example IfN all natural numbers 1 2 3 4 5 one useful partition on N is to consider E the even natural numbers and O the odd natural numbers E and 0 taken together and merged are N each element of N is in one or the other and E and O are disjoint We ll be doing a little work with sets in this module Triangles A triangle is the simplest of polygons You saw how one comes into existence with A5 We can talk about all triangles and prove theorems about all triangles For example in the preceding section we looked at a theorem that states that the sum of the interior angles of a triangle is 180 We can Visualize the collection of all triangles with an ellipse and imagine that all the triangles in the whole world are jumbled up inside The Set of All Triangles Once we ve Visualized the set it becomes important to organize it a bit Triangles are often categorized by side lengths A scalene triangle has three different side lengths The angle opposite the longest side is always the largest in a scalene triangle An isosceles triangle has two side lengths the same The base angles are always congruent in an isosceles triangle An equilateral triangle has all three side lengths the same Each angle measures 60 in an equilateral triangle Note then that all equilateral triangles are isosceles triangles but not all isosceles triangles are equilateral It is customary to designate an equilateral triangle by the more precise name if you know it applies rather than the looser label isosceles In order to organize all triangles I will distinguish between equilateral triangles and those that are purely isosceles ie have two congruent side lengths and a base that is different in length Here is a picture of the set of all triangles organized this way purely isosceles triangles with scalene base length triangles different from equilateral s1des triangles This is the whole set of all triangles divided into disjoint subsets Which is to say that each triangle ts into one and only one subset and if we combined the three subsets back to one set we d recover the original set of all triangles So we ve tidied up the set of all triangles considerably by doing this There is another big set of triangles that are important right triangles And it s not quite possible to see them in the organizational chart above We need to highlight right triangles and talk about how they t in ht triangles equilateral triangles scalene triangles purely isosceles Note that the subset right triangles actually crosses two of the disjoint subsets So right triangles is not disjoint from either scalene triangles or purely isosceles triangles it is a subset of both Right Triangles Exercise Sketch an isosceles right triangle What fact is always true about the base angles of an isosceles right triangle What fact is always true about the hypotenuse of an isosceles right triangle Does this help distinguish between isosceles right triangles and all the other isosceles triangles Sketch a scalene right triangle What is true that about the other two angles in the triangle the ones that are not the right Doe this fact help distinguish scalene right triangles from all the other scalene triangles Why are the right triangles disjoint from the equilateral triangles Triangles can be categorized by angle measure rather than side length Acute triangles have all three angles are acute Obtuse triangles have one obtuse angle Right triangles have a right angle Angle Measure Subsets Exercise Sketch the set of all triangles organized by angle measure If we group triangles like this are there any triangles in two groups or more Are the subsets disjoint Triangles and Angle Measure Theorem Exercise Prove that a triangle can have at most one angle measuring 90 or more Right Triangle Proof Exercise Given GR RU RP R0 and GP OU Prove AGRP is a right triangle P U 1 2 Areas and altitudes of similar triangles If you have two triangles it is an interesting problem to see if the triangles are related in any way Here is the way to determine if the triangles are related Establish whether or not the triangles both fall into one of the basic sets scalene purely isosceles with a base length different from the sides or equilateral OR acute obtuse right If the triangles fall into different sets then they are not related except that they are both triangles If both triangles fall into the same category then see if they are congruent using one of the congruence criteria SAS Postulate ASA Theorem AAS Theorem SSS Theorem If they are not congruent we have one more relationship we can try we can see if the triangles are similar Similarity between two triangles is denoted N If AABC N ADFE then the following six facts are true A A is congruent to A D A B is congruent to A F A C is congruent to A E and there is a positive number k the constant of proportionality such that AB k DF CBkFE ACkDE We summarize these siX facts by saying the angles of the two triangles are congruent and the side lengths are proportional Note that the naming convention says that you line up the corresponding angles in the same order in BOTH triangles AABC N ADFE does NOT tell you that angle A is congruent to angle F it DOES tell you that angle A is congruent to angle D There are some very nice ratios we can build using these facts Note that AB CB AC DFFEDE We will use this fact a bit in the future Similarity and Congruence in Triangles Exercise If two triangles are congruent are they also similar What is the scale factor in this case If two triangles are similar are they also congruent Let s put this in a subset relationship Is it similar triangles with a subset of congruent triangles or the other way around Sketch the larger set and the subset here Congruent Triangles Exercise Given M is the midpoint of AABC is isosceles Prove AACM E A BCM C A B There is a wonderful theorem that allows a short cut for checking similarity called the AA Similarity Theorem If two angles of one triangle are congruent to two angle of a second triangle then the triangles are similar There are many subsets of the set of all triangles that have the property that all triangles in the subset are similar Can you see why Email me if you don t see it o All isosceles right triangles are similar 0 All 3 7 4 7 5 triangles are similar 0 All equilateral triangles are similar We have a postulate that stipulates that congruent triangles have the same area A18 There is also a relationship between the area of a given triangle and one that is similar to it Let s work through a couple of exercises and come up with a conjecture about this relationship Areas and Altitudes of similar triangles exercise A 3 7 4 7 5 triangles In the following 3 7 4 7 5 right triangle the height is 3 units In a similar triangle on the right the height is 15 the scale factor or constant of proportionality from left to right is 12 4 2 What is the area of each triangle What is the ratio of the area of the smaller triangle on the right to the area of the triangle on the left What is the constant of proportionality or scale factor How are these numbers related What is the altitude of each triangle What is the ratio of the smaller to the larger How is this related to the constant of proportionality B Isosceles right triangles Here are two similar of course isosceles right triangles A39A 3393 What is the constant of proportionality from left to right What is the area of each triangle What is the ratio of the area of the larger triangle on the right to the area of the triangle on the left How are these numbers related What is the altitude of each triangle What is the ratio of the smaller to the larger How is this related to the constant of proportionality C Here is a pair of arbitrary similar triangles First Make a conjecture about the relationship of the altitudes and the areas What do you think is going to be the relationship DD39 600 cm AA39 400 0m E39D 300 cm AB39 200 0m E39F 260 cm 8390 173 cm is the altitude of AAA B and E39F is the altitude of ADD E Why do you suppose we use the word height in the area formula when we really mean length of the altitude email me or use the discussion board What is the scale factor k from AAA B to ADD E What are the areas What is the ratio of the area of AAA B to ADD E What about the ratio of the altitudes Summary of area and altitude of similar triangles exercise How do you know the triangles are similar What is the constant of proportionality B B D D C A C A39 BA 350 cm BquotA39 525 cm AC 250 cm sz A39Cquot135ooo szAC13500 MB C A 32460 mzACB 32460 Calculate the exact area of the smaller triangle using the trigonometric formula What is the area of the triangle on the right 7 without calculating What is the length of A D if AD p cm Here s a proof of what we ve gured out about altitudes Notice how the proof is related to our work on the exercises but is abstracted from any particular pair of triangles It is the abstraction process that makes the theorem applicable to any pair of similar triangles Theorem and proof If AABC AXYZ with a scale factor ofk and E is an altitude of AABC while is a corresponding altitude of AXYZ then CD kZW Case 1 The altitude is interior to the triangle Z C A D B X W Y 1 AABC N AXYZ 1 Given de nition of altitude CD J AB W J W 2 mzA sz 2 De nition similar 3 mzCDA mzZWX 3 Altitudes are perpendicular to the base each forms a 900 angle 4 AACD N AXZW 4 AA Similarity Theorem 5 AC kXZ 5 Given 6 CD kZW 6 CPCF QED Case 2 The altitude is exterior to the triangle 1 AABCNAXYZ lAB l 2 m AACB m AXZY 3 A ACD is supplementary to A ACB 4 A XZW is supplementary to A XZY 5 A ACD is supplementary to A XZY 6 A ACD E A XZW 7 AADC E AXWZ 8 AACD N AXWZ 9 AC kXZ 10 AD kXW QED Given de nition of altitude N De nition of similar E The Supplement Postulate A14 4 The Supplement Postulate A14 V39 substitution 0 Theorem supplements to the same angle gt1 de nition altitude they re both 900 9 AA Similarity Theorem 50 given 10 de nition similar So altitudes just get the same scale factor as sides Now let s do an analysis of what we ve gured out about areas of similar triangles Let s look at the formula for the area of a triangle and then analyze the formula for the area of a similar triangle where the scale factor from the first to the second is 3 Area of the first triangle is 12 baseheight What is the area of the second triangle 12 3base 3hei ht 12 9 base hei ht g g What is the ratio of the transformed triangle to the initial triangle 93218 Does this help explain what you observed I sure hope so Does it also work when you re using the trigonometric definition of area Area initial triangle ab sin 9 Area ofthe transformed triangle kakb sin 9 kzab sin 9 Sure it does Recognizing similarity We will continue working with similar triangles by working on spotting similar triangles in some constructions Example Constructing a congruent angle and similar triangles with any angle Prove that AADG N AFEG and that 4A E 4F Note Figure CAB is an angle So the construction is Take a perpendicular line to one leg ofan angle and take a perpendicular to the second leg 7 extend them until they intersect This construction creates similar triangles Proof ADGA E AEGF because they re vertical angles AADG E AGEF by construction So by the AA Similarity Theorem AADG N AFEG And in particular note that angle A is congruent to angle F Note the alternate illustration 7 it doesn t change the substance of the proof Similarity Proof Exercise Constructing similar triangles with parallel lines Prove that AABC N AEBD You are given that AC is parallel to DE If 7C g E what do you know about the area of AABC relative to the area of AEBD The construction is to take parallel lines and cross them with two transversals that intersect inbetween the parallel lines Both of these constructions give you a quick way to construct an angle that is congruent to a given angle or to construct similar triangles Exercise Constructing similar isosceles triangles An easy way to construct a pair of nested similar isosceles triangles follows Take an isosceles triangle in which the base is not equal to the sides and the apex angle is less than 60 Using the base as a side construct a smaller interior triangle inside the original one by rotating a ray up from the base until the new angle created inside one of the base angles is congruent to the apex angle of the original triangle mZBAC 35000 Prove that AABC N ABDC The two triangles share A C By construction 4A 5 ADBC D So by the AA Similarity Theorem AABC NABDC szBC 35000 If E is 6 cm and Eis 2 cm how long is E What is the scale factor from AABC to ABCD k 13 sum 03 3 3 How do the areas compare Area ABCD 19 area AABC 19 12 66 sin35 exactly 20 Exercise Constructing similar right triangles with an interior altitude Three similar right triangles are created by the altitude from the right angle to the hypotenuse of any right triangle We will use this construction at the end of this section in a proof of the Pythagorean Theorem AACD N AABD They share angle A and both have B one right angle AA Similarity mgADB 90000 ACBD N ADBA They share angle B and both have one right angle AA Similarity mgADB 90000 21 AACD N ADCB Which is to say the two smaller triangles are also similar Note that A A and A B are complements A A 900 7 A B In AACD we have a right angle A A and AADC90 4A 4B mgADB 90000 In ADCB we have a right angle and A B So by the AA Similarity Theorem these are similar Thus by constructing an interior altitude you get 3 pairs of similar right triangles Also note that the following ratios hold m4 Q hypotenuse BD CB shortest leg E hypotenuse AD AC medium leg 22 Another handy way to build similar triangles or to recognize when you are dealing with them is to note that Theorem Triangles that are nested and that share one vertex while having parallel sides across from that vertex are similar Given AABC nested into AADE with Eparallel to E demonstrate why AABC NAADE Since parallel to E AABC E AADE since they are corresponding angles and since the triangles share angle A the AA Similarity Theorem says they re similar Another nested construction 23 Nested triangles with no shared vertices paIt of a side shared and two sides parallel are similar Given E parallel to E DE parallel to AB We have then AFDE E ACAB since AC is atransversal and these angles are corresponding A similar argument gives ADFE E AACB And the AA Similarity Theorem provides that they are similar And one with parallel lines Triangles that have three sides parallel are similar Let s trace through the 3 angles labeled 1 They are congruent Because E is parallel to E the two angles labeled 1 and la are corresponding Because CB is parallel to FE the third angle labeled lb is corresponding to la Thus they are all congruent By a similar argument get a second angle congruent and apply the AA Similarity Theorem 24 Similar Triangles nested Exercise Given AOEV is isosceles with OE EV AABV is isosceles with BA BV m E m4 1 Prove AOEV N AABV O B E A 25 Inequality Theorems for Triangles First a review of some Arithmetic Properties for nonzero real numbers a b and c APl AP3 AP4 Ifaltbandbltcthenaltc This is called the transitivity property of inequalities To illustrate 5 lt 1 and 1 lt 7 so 5 lt 7 Ifabcthenagtbandagtc This says the sum of two numbers is bigger than either of the summands that were added together Addition of any number and multiplication by a positive number leaves an inequality unchanged Multiplication by a negative number reverses the sense of an inequality An illustration For X lt 5 X a lt 5 a and aX lt 5a as long as a gt 0 But the statement 7 X lt 5 means that X gt 7 5 The trichotomy law a lt b a b or a gt b The trichotomy law for real numbers a and b says that if you have two real numbers and you compare them then one is bigger than the other or they re both equal a lt b a b or a gt b There are no other options than these for comparing numbers 26 Let s jump right into a proof using these facts Theorem The measure of an exterior angle is greater than that of either remote interior angle Illustration The angle labeled Exterior Angle is bigger than 4A or AC Exterior Proof Let s start by stating that m A A m A C m A CBA 1800 by an earlier theorem Now since A CBA and the Exterior Angle are a linear pair they re supplementary by axiom m A CBA m 4 Exterior 180 Now subtract the second equation from the first You will nd that mzAmzCiszxterior0 so m 4 Exterior m A A m A C And since angles measures are just numbers between 0 and 180 we know that the sum is greater than either summand 2 above Arithmetic Properties Thus the Exterior Angle is larger than either remote interior angle Moving along The transitive nature of inequalities is used in the next proof and the Tricotomy Law gets used in the following proof Our strategy with the Tricotomy Law is to show that a given side is not shorter nor equal to another side Thus it must be longer This is not a comfortable strategy for beginners it s called an elimination proof you eliminate two possibilities so only the third remains 27 Unequal Sides and Angles Theorem and proof If the measures of two sides of a triangle are not equal then the measures of the angles opposite those sides are unequal in the same order Given AB gt AC Show mzACB gt m B B C We have that side AB gt side AC Locate point D on side AB so that you can construct a segment on the interior of the triangle AD so that AD AC thus creating an isosceles AADC Note that point D is in the interior of A ACB Now the base angles of AADC are congruent By Axiom 13 we have that mzACB mAACD szCB From arithmetic then mzACB is bigger than either summand in particular mzACB gt mzACD mzADC gt sz as desired because 4 ADC an exterior angle to ABDC and is thus larger than the remote interior angle B Unequal Sides Exercise List the angles of this triangle in order largest to smallest AC 28 Unequal Angles and Sides Theorem and proof If the measures of two angles of a triangle are unequal then the lengths of the sides opposite these angles are unequal in the same order Given m B gt mzC Show AC gt AB Let s suppose AB gt AC Then by the preceeding theorem mzC gt sz This is impossible because of what we are given Ok then let s suppose AB AC then sz mzC because they d be the base angles of an isosceles triangle Which isn t possible because we re given that angle B is larger Ok now let s review AB is NOT larger than AC and it s NOT equal to AC And these measurements are numbers so the only remaining possibility is AB lt AC by AP4 the Tricotomy Law Unequal Angles Exercise List the sides lengths is order largest to smallest for this triangle 29 The Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side The justi cations for this proof are a homework assignment 30 Pythagorean Theorem In a right triangle the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse AABC is a right triangle with altitude CD a c b aX bC X h otenuse h otenuse we are us1ng the ratios yp and yp medium leg shortest leg Working with the equalities we have a2b2 cxc2 cxc2 This proves the theorem 31 Convex Quadrilaterals A polygon with four sides is called a quadrilateral Here is a convex quadrilateral known as an arbitrary convex quadrilateral because all four side lengths are different Convex quadrilaterals are a proper subset of the set of all quadrilaterals What would be a good de nition of an exterior angle of a quadrilateral What is the sum of the exterior angles of this quadrilateral Do you know this is true for all quadrilaterals 32 How would you calculate the area of an arbitrary convex quadrilateral like this one Hint A19 We ll connect B and D decomposing the quadrilateral into two triangles then construct altitudes from A and C to the base line Then we ll calculate the area of each triangle and add them per A19 A hlb hzb hl 112 This is the area formula for the arbitrary convex quadrilateral 33 Just as with triangles there are many kinds of quadrilaterals We will olTer only one set of de nitions with its concomitant set containment structure One subset of the convex quadrilaterals is trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides called the bases b and b The height h is the perpendicular distance from b to b height h h base b C ABCD is a trapezoid Are trapezoids convex Yes they inherit this property because they are a subset of the convex quadrilaterals In order to nd a formula for the area of trapezoid ABCD we can take the trapezoid and recast it as two triangles each with height h by constructing diagonal BD Sketch in BD on the picture above Do you see the two triangles AABD and ABCD Invoking A19 we say that the area of trapezoid ABCD the area of AABD the area of ABDC By substitution we have that area ofthe trapezoid ABCD 12 b h 12 bh Since 12 and h are common factors we can factor them out to get our formula area oftrapezoid ABCD 12 b b h This is not much simpler than the formula for the arbitrary convex quadrilateral BUT we ve gotten quite a bit more specific about the properties of our quadrilateral You will see that the simplest area formula goes to the quadrilateral with the longest most complicated most restrictive set of properties 34 So we have all the convex quadrilaterals and we have a proper subset of some of them called the trapezoids Convex Quadrilaterals trapezoids The next restriction that is imposed on quadrilaterals is to have exactly two pairs of parallel sides This type of quadrilateral is a parallelogram In some textbooks the de nitions allow parallelograms to be a subset of trapezoid what we have chosen is that parallelograms and trapezoids are disjoint sets convex quadrilaterals trapezoids parallelo grams A parallelogram is a quadrilateral with two pairs of parallel sides 35 Here s an arbitrary parallelogram We have that Eis parallel to E and is parallel to Consider a pair of parallel sides and regard one of the opposite sides as a transversal Using the extensions of the sides argue that adjacent angles of the parallelogram are supplementary This gives us the theorem Adjacent angles of a parallelogram are supplementary You will prove this in your homework Theorems Opposite angles of a parallelogram are congruent Opposite sides of a parallelogram are congruent You may use these without proving them 36 Tests for Parallelograms Theorems If both pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram If both pairs of opposite angles of a quadrilateral are congruent then the quadrilateral is a parallelogram If two opposite sides of a quadrilateral are congruent and parallel then the quadrilateral is a parallelogram Note that these are not iff theorems Area of a parallelogram is simpler than that of a trapezoid Why is it true that the area of AABD the area of ABDC the area of the parallelogram ABCD Axiom 19 What do we know about the relationship between AABD and ABDC They re congruent same base length and height Which axiom guarantees that congruent triangles have the same area What is area AABD We ll apply Axiom 19 again and nd that the area of a parallelogram base length height bh 37 This is certainly simpler than the formula we started with for an arbitrary convex quadrilateral We can use what we have learned so far to nd out an interesting fact about triangles Midpoint Connector Theorem Exercise The segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one half that of the base We are given AABC with D the midpoint of EB and E the midpoint of E is the midpoint connector segment mentioned in the hypothesis We may extend DE to a point F on the extension so that DE 5 EF We may connect F and C with a segment D We are to show that segment DE is parallel to base BC and DE 12 BC 38 Proof Fill in the blanks 1 N 4 V39 0 gt1 9 0 O D is the midpoint of E E is the midpoint of AC Ra and 52 s AADEEACFE Es s 43544 E is parallel to E D DFCB is a parallelogram is parallel to E s soDFBc DFDEEF DFDEDE 2DE BC QED 2 3 Vertical angles are congruent 4 by construction 5 6 CPCF 7 The Transitive Law see step 2 8 9 Opposite interior angles are congruent 10 Opposite sides are congruent and parallel 1 1 12 13 de nition between 14 Substitution law 15 Addition 16 17 Division of an equation 39 Theorem The diagonals of a parallelogram bisect on another To get this sketch we constructed diagonal E and used CPCF to nd 42 z6md43541 We constructed diagonal Rand used CPCF to nd 41545md48 z4 Now we can say that AAPD E ACPB by ASA This means that E E 6 and The rst congruence means that P is the midpoint of R and similarly P is the midpoint of Thus the two diagonals bisect on another 40 Rhombuses or Rhombi A rhombus is a parallelogram with adjacent sides congruent Here is a picture of the relationship parallelo grams Fill in the blanks in the following statement using the words parallelogram and rhombuses Rhombus Exercise All are but not all Since rhombuses are a subset of parallelograms the same theorems for parallelograms hold true for them Diagonals of a rhombus bisect one another because all rhombuses are parallelograms What other facts do you know about rhombuses Check the Tests for Parallelograms Theorems for ideas 41 There is another fact that distinguishes rhombuses from an arbitrary parallelograms Look at a couple of illustrations and make a conjecture H B B AI H A C rhombus B AB 03 cm H39H 352 cm BBI 300 cm HH39 352 cm parallelogram A I one that is nota rhombus C Cquot H C rhombus A CC39 300 cm parallelogram B C39Cquot 300 cm one that is not a rhombus DE 495 cm DD39 250 cm E D D What is going on with the diagonals that distinguishes the two rhombuses from the two parallelograms What is your conjecture 42 Theorem and proof The diagonals of a rhombus are perpendicular You are given that DABCD is a rhombus Prove that the diagonals are perpendicular 1 DABCD is a rhombus 1 Given 2 E E E 2 De nition rhombus 3 E E E 3 Diagonals of a parallelogram bisect each other 4 E E E 4 Re exive Law 5 AAED E AAEB 5 SSS Congruence Theorem 6 41542 6 CPCF 7 A 1 is adjacent to A 2 they re a linear pair 7 De nition of adjacent angles 8 E J E 8 De nition of perpendicular lines QED 43 Now we will move on to another quadrilateral RECTANGLES A rectangle is a parallelogram with one right angle All rectangles are parallelograms so the following properties hold Opposite sides and opposite angles are congruent Adjacent angles are supplementary The diagonals bisect one another The area ofa rectangle is height times base Rectangles Exercise All rectangles are similar true or false and why Theorem All angles of a rectangle are right angles This is to be proved in the homework 44 Theorem The diagonals of a rectangle are congruent Proof Let D ABCD be a rectangle and let A A be the right angle from the de nition We will prove that AC E BD Because opposite sides of a parallelogram are congruent we know that E E E By the Re exive law we know DC 5 DC A D E A C because they both measure 90 by the preceding theorem This means that AADC E A BCD by SAS Thus by CPCF R E E Note that we ve found two triangles in the rectangle AADC and ABCD This is a partial and overlapping decomposition You re allowed to do this There are two more 7 upside down 7 triangles that we might need someday too Do you see them AACB and AADB 45 We will explore some properties of similar rectangles What is the de nition of similar rectangles all rectangles have interior angles that measure 90 So we re really looking at a scale factor k The sides are proportional Similar Rectangles Exercise DABCD N DA B C D DA B C D and AB A B l5A B Calculate the ratio of height to base and the area for each of the rectangles A D39A39 200 cm DA 400 cm D39C39 300 cm DC 600 cm B D C39 Aquot Bquot DquotAquot 600 cm DquotCquot 900 cm Dquot Cquot Make a conjecture about the height to base ratios of similar triangles and 46 make a conjecture about the area of similar triangles In summary for similar triangles Similar triangles have the same height to base ratio The areas are related by a multiplier of k2 Similar Rectangles Area Exercise Look at the formula for area and see if you can gure out why k2 shows up when we take ratios 47 Now we will explore the quadrilateral with the most restricted de nition a Square A square is a rhombus with one right angle Sguare Subset Exercise Sketch in the picture of how rectangles squares and rhombuses are related parallelo grams What is the formula for the area of a square Is this the same as saying base times height What is the ratio of base to height for all squares True or false All squares are similar 48 Circles De nition of a circle A circle is the set of all points in the plane with a constant distance from a given point called the center of the circle Note that the circle divides the plane into 3 regions just like a triangle does Is a circle convex Area ofa circle 11r2 Circumference of a circle ie perimeter 1rd or 2r11 Proposition All circles are similar How might we change our de nition of similar to accommodate such a statement Since circles aren t polygon and don t have interior angles we might have to focus on the radius and the center If you take a center of one circle and it s radius is the radius of a second circle just a multiplier different from the rst What is the proportionality constant from circle A to circle B This is a rather nonstandard concept I just want to stretch your mind a bit De nitions Central angle 7 an angle with its vertex at the center of the circle AACB Inscribed angle 7 an angle with its vertex on a circle point zFED The measure of an arc is the number of degrees in the central angle that intercepts the arc 1MB mzACB 49 Theorem and proof Inscribed angle theorem The measure of an inscribed angle is one half the measure of its intercepted are There are 3 cases to consider one with the center of the circle on the angle side one with the center of the circle in the interior of the angle and one with the center of the circle exterior to the inscribed angle We start with the case with the center of the circle on the angle side ie one side of the inscribed angle is a diameter of the circle Given the center of the circle C is on the side of inscribed V A AVB Prove mzAVB 12 m arcAB Construct radius CA B Recall that by the de nition of central angle mzACB m arc AB By de nition radius AC 5 CP so AVCA is isosceles thus mzA sz also known as AAVB By the Exterior Angle Theorem mzACB mzAVB mzA By substitution we have mzACB ZmAAVB This means that mzAVB 12 mzACB 12 m arc AB as desired Case 2 Note that the center is in the interior of the angle we want to measure Run a diameter through C and V Apply case one to both A angles and add them V mAAVB szVC mzCVB 50 Case 3 Run a diameter through C and V Note that B is in the interior of angle AVC mzAVB mzAVC szVC A V B Corollary t0 the theorem Every angle inscribed in a semicircle is a right angle What is the measure of the intercepted arc 180 By the Inscribed Angle Theorem then what is the measure of angle ABC 51 Congruent Inscribed Angles Theorem Inscribed angles that intercept the same or equal arcs are congruent Inscribed Angles Exercise Prove that HC 5 FB I szCB 3800O mzHBC 38000 Tangent Lines to a circle A line that is tangent to a circle has a one point intersection with the circle Theorem A tangent line to a circle is perpendicular to the radius that shares with point of tangency 52 Theorem and proof Two tangent segments to a circle from the same point have equal lengths What are the givens What is to be proved Note that OA and OB are radii 7 what do we know then And too the radii are perpendicular to the tangent lines 7 from the fact above 7 what do we know about AAOP and ABOP More than just being right triangles 7 they re congruent right triangles why See answers to exercises Congruent right triangles Thus side AP is congruent to side BP which is what we re trying to prove 53 between two circles exercise Two circles may intersect in zero one or two points Draw these relationships zero one two You should have 4 sketches Tangent circles intersect at one point Theorem If two circles are tangent internally or externally the point of tangency is on their line of centers Line of centers exercise illustrate the above theorem see the solution to the above exercise 54 Some additional vocabulary Two circles may have a common external tangent line this happens when the circles are on the same side of the line Corrmon external langent line Two circles may have a common internal tangent line this happens when the circles are on opposite sides ofthe line Corrmon internal langent line 55 Answers and hints to selected exercises Right Triangles Exercise A Sketch an isosceles right triangle Use Sketchpad What fact is always true about the base angles of an isosceles right triangle They re always equal to 45 What fact is always true about the hypotenuse of an isosceles right triangle It s the square root of two times the length of the congruent sides lenth Does this help distinguish between isosceles right triangles and all the other isosceles triangles Yes if an isosceles triangle meets either of these conditions it is a right triangle The others don t have these properties B Sketch a scalene right triangle Use Sketchpad A 3 7 4 7 5 is one What is true that about the other two angles in the triangle the ones that are not the right angle They re unequal and complementary Does this fact help distinguish scalene right triangles from all the other scalene triangles Yes they ll be unequal in the other scalenes but not complementary C Why are the right triangles disjoint from the equilateral triangles The angle measures of the equilaterals is 60 each angle They can t be right triangles and right triangles can t have 3 60 angles Angle Measure Subsets Exercise Yes the subsets are disjoint obtuse triangles acute triangles right triangles 56 Triangles and Angle Measure Theorem Exercise Suppose we have a triangle that has one angle of 90 Then since the sum of the angles of a triangle is 180 The other two angles are acute and complementary This argues that a triangle may have at least one angle that measures 90 Now suppose the triangle has one angle that measures more than 90 suppose it measure 90 X where 0 lt X lt 180 per our axioms Then since the sum ofthe angles ofa triangle is 180 the other two angles add to 180 7 X Which is ne This argues that a triangle may have at least one angle that is obtuse Now suppose the triangle has two right angles two obtuse angles or one right and one obtuse This can t happen because then the sum of the other two angles has to be zero or negative and angles can t have these measures This argues that there is eXactly one and not more than one angle 90 or bigger in a triangle Right Triangle Proof EXercise P U Given GR RU RP R0 and GP OU Prove AGRP is a right triangle 1 2 The two triangles are congruent by SSS G O Angles l and 2 are congruent by CPCF and since they re a linear pair and congruent they re a supplementary pair and they each measure 90 Thus BOTH triangles are right triangles and in particular AGRP is right 57 Similarity and Congruence in Triangles Exercise If two triangles are congruent are they also similar What is the scale factor in this case Yes they are the constant of proportionality aka scale factor is 1 If two triangles are similar are they also congruent No they re not necessarity unless the scale factor is 1 Let s put this in a subset relationship Is it similar triangles with a subset of congruent triangles or the other way around All congruent triangles are similar Not all similar triangles are congruent just some few Similar triangles Congruent triangles Congruent triangles is a proper subset of Similar triangles Congruent Triangles Exercise Given M is the midpoint of C AABC is isosceles Prove AACM E A BCM Since M is the midpoint AM MB Since the triangle is isosceles AC CB Since CM is shared the two smaller triangles are congruent by SSS or You may use the fact that base angles of the I isosceles triangle are congruent and go for A M B a SAS argument 58 Areas and Altit udes of Similar Triangles exercise A Left area is 6 right area is 32 2 Ratio32lll 18 124 2 Left altitude is 3 right altitude is k3 32 B k2 Left area is 2 right area is 8 Ratio E 4 k2 2 Left altitude is 2 right altitude is k2 4 C conjecture area transformed k2 area original Summary of ar altitude transformed kaltitude original k 32 ratio 94 altitude right 32 altitude left ea and altitude of similar triangles exercise AA similarity Theorem 32 Area smaller gab sin 9 3 52 5 sin 135 35x5 3 Area lar er 2 g 2 16 Altitude larger 3 59 Similarity Proof Exercise Prove that AABC N AEBD You are given that AC is parallel to DE Proof Note that all three interior angles of AACB are congruent to all three interior angles of AEBD Regarding Ali as atransversal ACAB E ADEB Regarding 613 as atransversal ABCE E ABCA And because vertical angles are congruent AABC E ADBE So by the AA Similarity theorem the triangles are similar If C i g E the area of AABC 925 the area of AEDB 60 Similar Triangles nested Exercise 0 Given AOEV is isosceles with OE EV AABV is isosceles with BA BV m E m4 1 Prove AOEV N AABV 3 Since AOEV is isosceles A O and A V are congruent Since AABV is isosceles A V and A BAV are congruent Thus the base angles of the two triangles are all 3 congruent Applying the AA Similarity Theorem the triangles are congruent Unequal Sides Exercise 4C gt AB gt LA Unequal Angles Exercise ABgtACgtCB Midpoint Connector Theorem Exercise The segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one half that of the base We are given AABC with D the midpoint of E and E the midpoint of E is the midpoint connector segment mentioned in the hypothesis We may extend DE to a point F on the extension so that DE 5 EF We may connect F and C with a segment 61 Proof 7 Fill in the blanks l N E 4 V39 0 gt1 9 0 O 17 D is the midpoint of E E is the midpoint of AC REE and Es mzlmz2 z AADEEACFE Ez s 43544 E is parallel to E D DFCB is a parallelogram is parallel to E sEsoDFBC DFDEEF DFDEDE DF2DE 2DE BC DE12BC QED l Given 2 De nition midpoint 3 Vertical angles are congruent 4 by construction 5 SAS 6 CPCF 7 The Transitive Law see step 2 8 CPCF 9 Opposite interior angles are congruent 10 Opposite sides are congruent and parallel 11 De nition parallel 12 De nition congruence l3 de nition between 14 Substitution law 1 5 Addition l6 Substitution from step 12 17 DiVision of an equation 62 Rhombus Exercise All rhombuses are parallelograms but not all parallelograms are rhombuses So rhombuses are a proper subset of parallelograms Rectangle Exercise All rectangles are similar true or false and why False Here are two that aren t similar there s no k AA AA BBquot BB Similar Rectangles Exercise DABCD N DA B C D DA B C D and AB A B l5A B height to base ratio on all three is 23 Area on ABCD is 24 Area on A B C D is 6 less by k2 14 Area on A B C D is 54 more by k2 94 Conjecture area changes by k2 Height to base ratio is a constant for all similar rectangles Similar Rectangles Area Exercise Look at the formula for area and see if you can gure out why k2 shows up when we take ratios Area lw Area of a similar rectangle klkw hence the k2 63 Square Subset Exercise Sketch in the picture of how rectangles squares and rhombuses are related parallelo grams rectangle Squares are a proper subset of both rhombuses and rectangles They are in the intersection of both sets A square can be called a special rhombus or a special rectangle Inscribed Angles Exercise Prove that HC 5 FB szCB 3800O mzHBC 38000 B C Look at AHCB CB is shared with AFCB mzH sz because they subtend the same arcSo AHCB E AFCB by AAS Thus HC 5 FB by CPCF 64 Congruent right triangles Triangle AOP has 2 2 2 0A AP OP Triangle BOP has 2 2 2 OB BP OP Subtract these two 2 2 AP BP 0 E i but the negative is extraneous 7 distance can never be negative done Relationships between two circles exercise Two circles may intersect in zero one or two points Draw these relationships 90 one two 90 You should have 4 sketches Most people don t consider the internally tangent case but you need to Do you see the line of centers 65 Appendix A Theorems T1 The angle opposite the longest side is always the largest in a scalene triangle T2 The base angles are always congruent in an isosceles triangle T3 Each angle measures 60 in an equilateral triangle T4 A triangle can have at most one angle measuring 90 or more AA Similarity Theorem If two angles of one triangle are congruent to two angle of a second triangle then the triangles are similar Similar altitudes theorem If AABC N AXYZ with a scale factor of k and E is an altitude of AABC while W is a corresponding altitude of AXYZ then CD kZW Pythagorean Theorem Given a right triangle the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse Nl Triangles that are nested and that share one vertex while having parallel sides across from that vertex are similar N2 Nested triangles with no shared vertices part of a side shared and two sides parallel are similar N3 Triangles that have three sides parallel are similar EA The measure of an exterior angle is greater than that of either remote interior angle For a b and c Real Numbers APl Ifaltbandbltcthenaltc This is called the transitivity property of inequalities To illustrate 5 lt l and l lt 7 so 5 lt 7 AP2 Ifabcthenagtbandagtc This says the sum of two numbers is bigger than either of the summands that were added together 66 AP3 Addition of any number and multiplication by a positive number leaves an inequality unchanged Multiplication by a negative number reverses the sense of an inequality An illustration For X lt 5 X a lt 5 a and aX lt 5a as long as a gt 0 But the statement 7 X lt 5 means that X gt 7 5 AP4 The trichotomy law a lt b a b or a gt b The trichotomy law for real numbers a and b says that if you have two real numbers and you compare them then one is bigger than the other or they re both equal a lt b a b or a gt b There are no other options than these for comparing numbers Unequal Sides and Angles theorem If the measures of two sides of a triangle are not equal then the measures of the angles opposite those sides are unequal in the same order Unequal Angles and Sides theorem If the measures of two angles of a triangle are unequal then the lengths of the sides opposite these angles are unequal in the same order The Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side Parallelogram l Adjacent angles of a parallelogram are supplementary Parallelogram 2 Opposite angles of a parallelogram are congruent Parallelogram 3 Opposite sides of a parallelogram are congruent Parallelogram 4 If both pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram Parallelogram 5 If both pairs of opposite angles of a quadrilateral are congruent then the quadrilateral is a parallelogram Parallelogram 6 If two opposite sides of a quadrilateral are congruent and parallel then the quadrilateral is a parallelogram 67 Midpoint Connector Theorem The segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is onehalf that of the base R1 All angles of a rectangle are right angles R2 The diagonals of a rectangle are congruent Inscribed angle theorem The measure of an inscribed angle is onehalf the measure of its intercepted arc Cl Every angle inscribed in a semicircle is a right angle Congruent Inscribed Angles Theorem Inscribed angles that intercept the same or equal arcs are congruent C2 Theorem A tangent line to a circle is perpendicular to the radius that shares with point of tangency Theorems from the Axioms section A8 A line or part ofa line can be named for any 2 points on it All 7 a Complements to the same angle are congruent All 7 b Vertical angles are congruent A14 The angle bisectors of a linear pair are perpendicular A15 7 a If two angles and the included side of one triangle are congruent to the corresponding two and included side of another triangle then the triangles are congruent A15 7 b If under some correspondence three sides of a triangle are congruent to the three sides of another triangle then the triangles are congruent 68 A157c A167a A167b A167c A1641 Al6ie A164 A167g If under some correspondence a pair of angles and the side adjacent to one of them are congruent to a corresponding pair of angles and adjacent side then the triangles are congruent Two lines are parallel if and only if a pair of alternate exterior angles around a transversal are congruent Two lines are parallel if and only if two interior angles on the same side of a transversal are supplements Two lines are parallel if and only if a pair of corresponding angles around a transversal are congruent Two lines are parallel if and only if alternate interior angles around a transversal are congruent If two lines are perpendicular to the same third line then the lines are parallel The sum of the interior angles of a triangle are 180 The sum of the measures of the remote interior angles of a triangle is the same as measure of the exterior angle across from them 69 Appendix B SMSG Postulates for Euclidean Geometry A1 A2 A3 A4 A5 A6 A7 A8 A9 Given any two distinct points there is exactly one line that contains them The Distance Postulate To every pair ofdistinct points there corresponds a unique positive number This number is called the distance between the two points The Ruler Postulate The points ofa line can be placed in a correspondence with the real numbers such that To every point ofthe line there corresponds exactly one real number B To every real number there corresponds exactly one point ofthe line and C The distance between two distinct points is the absolute value of the difference of the corresponding real numbers The Ruler Placement Postulate Given two points P and Q ofa line the coordinate system can be chosen in such a way that the coordinate ofP is zero and the coordinate on is positive A Every plane contains at least three non collinear points B Space contains at least four non coplanar points Iftwo points line in a plane then the line containing these points lies in the same plane Any three points lie in at least one plane and any three non collinear points lie in exactly one plane If two planes intersect then that intersection is a line The Plane Separation Postulate Given a line and a plane containing it the points ofthe plane that do not lie on the line form two sets such that A each of the sets is convex and B ifP is in one set and Q is in the other then segment PQ intersects the line 70 The Space Separation Postulate The points of space that do not line in a given plane form two sets such that A each ofthe sets is convex and B ifP is in one set and Q is in the other then the segment PQ intersects the plane The Angle Measurement Postulate To every angle there corresponds a real number between 0 and 180 The Angle Construction Postulate Let AB be a ray on the edge ofthe half plane H For every r between 0 and 180 there is exactly one ray AP with P in H such that m A PAB r The Angle Addition Postulate IfD is a point in the interior of BAC then szACszADmADAC The Supplement Postulate If two angles form a linear pair then they are supplementary The SAS Postulate Given an one to one correspondence between two triangles or between a triangle and itself Iftwo sides and the included angle ofthe rst triangle are congruent to the corresponding parts ofthe second triangle then the correspondence is a congruence The Parallel Postulate Through a given external point there is at most one line parallel to a given line To every polygonal region there corresponds a unique positive number called its area Iftwo triangles are congruent then the triangular regions have the same area Suppose that the region R is the union oftwo regions R1 and R2 Ile and R2 intersect at most in a nite number of segments and points then the area ofR is the sum ofthe areas ole and R2 The area ofa rectangle is the product ofthe length ofits base and the length ofits altitude The volume ofa rectangular parallelpiped is equal to the product ofthe length of its altitude and the area ofits base 71 Cavalieri s Principal Given two solids and a plane If for every plane that intersects the solids and is parallel to the given plane the two intersections determine regions that have the same area then the two solids have the same volume 72 Appendix C Glossary A An angle with a measure less than 90 is called an acute angle Adjacent angles are a pair of angles with a common vertex a common side and no interior points in common The nonadjacent angles on opposite sides of the transversal and on the exterior of the two lines are called alternate exterior angles The nonadjacent angles on opposite sides of the transversal but on the interior of the two lines are called alternate interior angles A line segment from the vertex of a triangle to the side opposite that vertex or an extension of that side is called an altitude of the triangle Triangles have 3 altitudes that are concurrent at a single point called the orthocenter It is not necessary that an altitude be part of the triangle s interior Any ray segment or line that divides an angle into two congruent angles is called an angle bisector Ray AS is the angle bisector of A TAB if and only if S is an interior point of the angle and szAS szAB are equal The angle bisectors of a triangle are concurrent at point called the incenter We may then speak ofthe points between A and B We will say that a point C is between A and B denoted A 7 C 7 B if and only if AC CB AB C We say that a point A and a point B are collinear if they are on the same line Two angles whose measures sum to 90 are called complementary angles Three or more lines that intersect in a single point are said to be concurrent at that point 73 Two angles A and B with the same measure are called congruent angles We will use the notation A A E A B Two polygons are said to be congruent polygons if corresponding angles are congruent and corresponding sides are congruent Two segments are said to be congruent segments if they have the same length The constant of proportionality is the multiplier used to go from one triangle s side lengths to a similar triangle s side lengths A convex set has the property that the points between any two points of it are also set elements Examples include the interior of an angle A circle however is not a convex set Convex polygons are polygons in which a segment connecting two points of the polygon is composed entirely of points from the interior of the polygon Congruent parts of congruent gures are congruent This is always true and we will use the acronym CPCF when we mean to say this in a proof The nonadjacent angles on the same side of the transversal and in corresponding locations with respect to the non transversal lines are called corresponding angles 39d d39 t B The cosine of angle B denoted cosB or cos B is the ratio w hypotenuse D Two sets are said to be disjoint sets if they have no set elements that are shared which is to say that the list of elements of one contains nothing in common with the list of elements of the other E An equilateral triangle has all three side lengths the same Each angle measures 60 in an equilateral triangle An exterior angle to triangle is created when one side is extended to a ray The adjacent triangle side at the extension point and the ray form an exterior angle to the triangle G 74 A geometric coordinate is a single real number specifying the location of a point on a line The absolute value of the difference of two geometric coordinates is the distance between them These are speci ed in Axiom 3 Composition of three re ections produces a glide re ection iff is a convenient way to compress two implications into a single statement If A then B and also if B then A becomes A iff B It is an abbreviation of if and only if A point that is inbetween the points of intersection of a transversal and the rays forming an angle is an interior point of the angle A point is an interior point of a polygon if it is between the intersection points of transversal and two sides or vertices of the polygon A transformation that preserves the distance between any two points is an isometry An isosceles triangle has two side lengths the same The base angles are always congruent in an isosceles triangle L The Law of Sines for arbitrary triangles given a triangle with vertices A B and C with sides opposite the respective vertices a b and c it is true that sinA sinB sinC a b c Adjacent angles whose nonshared rays form a straight line are called a linear pair M A segment from one vertex of a triangle to the midpoint of the side opposite is called a median Each triangle has 3 medians that are concurrent at a point called the centroid Given a segment E there is exactly one point C such that A 7 C 7 B and AC CB This point is called the midpoint of the segment 0 An angle with a measure greater than 90 is called an obtuse angle 75 P A parallelogram is a quadrilateral with two pairs of parallel sides A permutation is a reordering of a list of objects in a way that does not allow items on the list to jump For example A B C D is an ordered list D A B C is a permutation of the list while A D B C is not You must always move the last letter to the front of the list to get another permutation This is a special use of the word The perpendicular bisector of a segment is the collection of all points equidistant from the endpoints of a segment Each triangle has 3 perpendicular bisectors that are concurrent at a point called the circumcenter of the triangle Two lines making vertical angles with one of the angles measuring 90 are called perpendicular lines A polygon is a closed gure in a plane It has 3 or more segments called sides that intersect only at their endpoints Each point of intersection is called a vertex No two consecutive sides are on the same line If we say a quality is preserved or an operation preserves a quality we mean that the quality mentioned is not changed by the transformation it is invariant A solid gure formed byjoining two congruent polygonal shapes in parallel planes is called a prism The sides of a prism are parallelograms Proper subset see Subset The Pythagorean Theorem In a right triangle the sum of the squares of the legs is the same as the square of the hypotenuse In Trigonometry this quickly becomes the Pythagorean Identity sin2 639 cos2 639 1 R A re ection is an isometry that takes the original set and creates a mirror image of it across a speci ed line of re ection Polygons constructed from congruent segments and having congruent interior angles are called regular polygons A rhombus is a parallelogram with adjacent sides congruent An angle with a measure of 90 is called a right angle A rotation is the composition of two re ections across intersection lines of re ection 76 S A scalene triangle has three different side lengths The angle opposite the longest side is always the largest in a scalene triangle Given two triangles AA and AB we say AA is similar to AB denoted AA N AB if and only if corresponding angles are congruent and the measures of corresponding sides are in the same proportion A transformation that changes the distances between given points by a fixed amount called the scale factor is called a similarity transformation or a dilation A similarity transformation must have a point of dilation specified 39d 39t B The sine of angle B denoted s1nB or sm B 1s the ratlo w hypotenuse A subset S of a set A is a set that has all of its elements from the set A We denote a subset S of A this way S g A A proper subset P is one that is actually smaller in size than the original set A All of P s elements are also in A but A has at least one element that is not in P denoted P CA Two angles whose measure sum to 180 are called supplementary angles T 39d 39t B The tangent of angle B denoted tanB or tan B 1s the ratlo w s1de adjacent B Two sets are said to be tangent if they intersect or share exactly one point A translation is the composition of two re ections across parallel lines of re ection In geometry the process by which one can effect a change to a set ofpoints is called a transformation A transformation requires an original set and some specific instructions Any line that intersects two other lines or parts of lines is called a transversal of the lines rays or segments intersected A trapezoid is a quadrilateral with exactly one pair of parallel sides called the bases b and b The height h is the perpendicular distance from b to b The trichotomy law for real numbers a and b says that if you have two real numbers and you compare them then one is bigger than the other or they re both equal a lt b a b or a gt b There are no other options than these for comparing numbers 77 V Two angles that are not adjacent and yet are formed by the intersection of two straight lines are called vertical angles 78
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