In Exercises 1 and 2, copy and complete the statement. The ? of a trapezoid is parallel

VOCABULARY Sketch a convex hexagon. Draw all of its diagonals.

WRITING How many exterior angles are there in an n-gon? Are all the exterior angles considered when you use the Polygon Exterior Angles Theorem? Explain.

INTERIOR ANGLE SUMS Find the sum of the measures of the interior angles of the indicated convex polygon. Nonagon

INTERIOR ANGLE SUMS Find the sum of the measures of the interior angles of the indicated convex polygon. 14-gon

INTERIOR ANGLE SUMS Find the sum of the measures of the interior angles of the indicated convex polygon. 16-gon

INTERIOR ANGLE SUMS Find the sum of the measures of the interior angles of the indicated convex polygon. 20-gon

FINDING NUMBER OF SIDES The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides. 3608

FINDING NUMBER OF SIDES The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides. 7208

FINDING NUMBER OF SIDES The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides. 19808

FINDING NUMBER OF SIDES The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides. 23408

ALGEBRA Find the value of x. 868 1408 1388

ALGEBRA Find the value of x. x8 1628 1218 968 1018

ALGEBRA Find the value of x. x8 2x8 1438 1398 1528 1168 1258 140

ALGEBRA Find the value of x. x8 788 658 1068

ALGEBRA Find the value of x. 2x8 x8 45

ALGEBRA Find the value of x. 508 398 588 488 598

ERROR ANALYSIS A student claims that the sum of the measures of the exterior angles of an octagon is greater than the sum of the measures of the exterior angles of a hexagon. The student justifies this claim by saying that an octagon has two more sides than a hexagon. Describe and correct the error the student is making.

MULTIPLE CHOICE The measures of the interior angles of a quadrilateral are x8, 2x8, 3x8, and 4x8. What is the measure of the largest interior angle? A 1208 B 1448 C 1608 D 3608

REGULAR POLYGONS Find the measures of an interior angle and an exterior angle of the indicated regular polygon. Regular pentagon

REGULAR POLYGONS Find the measures of an interior angle and an exterior angle of the indicated regular polygon. Regular 18-gon

REGULAR POLYGONS Find the measures of an interior angle and an exterior angle of the indicated regular polygon. Regular 90-gon

DIAGONALS OF SIMILAR FIGURES Hexagons RSTUVW and JKLMNP are similar. }RU and }JM are diagonals. Given ST 5 6, KL 5 10, and RU 5 12, find JM. N M S K P L R

SHORT RESPONSE Explain why any two regular pentagons are similar.

REGULAR POLYGONS Find the value of n for each regular n-gon described. Each interior angle of the regular n-gon has a measure of 1568.

REGULAR POLYGONS Find the value of n for each regular n-gon described. Each exterior angle of the regular n-gon has a measure of 98.

POSSIBLE POLYGONS Determine if it is possible for a regular polygon to have an interior angle with the given angle measure. Explain your reasoning. a. 1658 b. 1718 c. 758 d. 408

CHALLENGE Sides are added to a convex polygon so that the sum of its interior angle measures is increased by 5408. How many sides are added to the polygon? Explain your reasoning.

BASEBALL The outline of the playing field at a baseball park is a polygon, as shown. Find the sum of the measures of the interior angles of the polygon.

BASEBALL The outline of the playing field at a baseball park is a polygon, as shown. Find the sum of the measures of the interior angles of the polygon.

JEWELRY BOX The base of a jewelry box is shaped like a regular hexagon. What is the measure of each interior angle of the hexagon?

GREENHOUSE The floor of the greenhouse shown is a shaped like a regular decagon. Find the measure of an interior angle of the regular decagon. Then find the measure of an exterior angle.

MULTI-STEP PROBLEM In pentagon PQRST, P, Q, and S are right angles, and R > T. a. Draw a Diagram Sketch pentagon PQRST. Mark the right angles and the congruent angles. b. Calculate Find the sum of the interior angle measures of PQRST. c. Calculate Find m R and m T.

PROVING THEOREM 8.1 FOR PENTAGONS The Polygon Interior Angles Theorem states that the sum of the measures of the interior angles of an n-gon is (n 2 2) p 1808. Write a paragraph proof of this theorem for the case when n 5 5. E D C B

PROVING A COROLLARY Write a paragraph proof of the Corollary to the Polygon Interior Angles Theorem.

PROVING THEOREM 8.2 Use the plan below to write a paragraph proof of the Polygon Exterior Angles Theorem. Plan for Proof In a convex n-gon, the sum of the measures of an interior angle and an adjacent exterior angle at any vertex is 1808. Multiply by n to get the sum of all such sums at each vertex. Then subtract the sum of the interior angles derived by using the Polygon Interior Angles Theorem.

MULTIPLE REPRESENTATIONS The formula for the measure of each interior angle in a regular polygon can be written in function notation. a. Writing a Function Write a function h(n), where n is the number of sides in a regular polygon and h(n) is the measure of any interior angle in the regular polygon. b. Using a Function Use the function from part (a) to find h(9). Then use the function to find n if h(n) 5 1508. c. Graphing a Function Graph the function from part (a) for n 5 3, 4, 5, 6, 7, and 8. Based on your graph, describe what happens to the value of h(n) as n increases. Explain your reasoning.

EXTENDED RESPONSE In a concave polygon, at least one interior angle measure is greater than 1808. For example, the measure of the shaded angle in the concave quadrilateral below is 2108. a. In the diagrams above, the interiors of a concave quadrilateral, pentagon, hexagon, and heptagon are divided into triangles. Make a table like the one in the Activity on page 506. For each of the polygons shown above, record the number of sides, the number of triangles, and the sum of the measures of the interior angles. b. Write an algebraic expression that you can use to find the sum of the measures of the interior angles of a concave polygon. Explain.

CHALLENGE Polygon ABCDEFGH is a regular octagon. Suppose sides AB}and }CD are extended to meet at a point P. Find m BPC. Explain your reasoning. Include a diagram with your answer.

Find m 1 and m 2. Explain your reasoning. (p. 154) 1 988 2

Find m 1 and m 2. Explain your reasoning. (p. 154) 2 15

Find m 1 and m 2. Explain your reasoning. (p. 154) 1 2 1268

Quadrilaterals JKLM and PQRS are similar. If JK 5 3.6 centimeters and PQ 5 1.2 centimeters, find the scale factor of JKLM to PQRS. (p. 372)

Quadrilaterals ABCD and EFGH are similar. The scale factor of ABCD to EFGH is 8 : 5, and the perimeter of ABCD is 90 feet. Find the perimeter of EFGH. (p. 372)

Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. (p. 483) sin A 5 0.77

Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. (p. 483) sin A 5 0.35

Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. (p. 483) cos A 5 0.81

Let A be an acute angle in a right triangle. Approximate the measure of A to the nearest tenth of a degree. (p. 483) cos A 5 0.23

VOCABULARY What property of a parallelogram is included in the definition of a parallelogram? What properties are described by the theorems in this lesson?

WRITING In parallelogram ABCD, m A 5 658. Explain how you would find the other angle measures of ~ABCD.

ALGEBRA Find the value of each variable in the parallelogram. 5 9 y x

ALGEBRA Find the value of each variable in the parallelogram. 2 n m

ALGEBRA Find the value of each variable in the parallelogram. 558 a8

ALGEBRA Find the value of each variable in the parallelogram. 2p8 1208

ALGEBRA Find the value of each variable in the parallelogram. d 2 21)8 2

ALGEBRA Find the value of each variable in the parallelogram. (g 1 4)8

FINDING ANGLE MEASURES Find the measure of the indicated angle in the parallelogram. Find m B A D 518

FINDING ANGLE MEASURES Find the measure of the indicated angle in the parallelogram. Find m L. N L P 958

FINDING ANGLE MEASURES Find the measure of the indicated angle in the parallelogram. Find m Y X Z Y 1198

SKETCHING In ~PQRS, m R is 24 degrees more than m S. Sketch ~PQRS. Find the measure of each interior angle. Then label each angle with its measure.

ALGEBRA Find the value of each variable in the parallelogram. 15 9 5a

ALGEBRA Find the value of each variable in the parallelogram. 16 2n 9 2 n 4m

ALGEBRA Find the value of each variable in the parallelogram. 5y 12 3x 4y 1 4

MULTIPLE CHOICE The diagonals of parallelogram OPQR intersect at point M. What are the coordinates of point M? A 11, 5 }22 B 12, 5 }22 C 11, 3 }22 D 12, 3 }22 6 1 Q M P O R

REASONING Use the photo to copy and complete the statement. Explain. A B D C 868 AD > ?

REASONING Use the photo to copy and complete the statement. Explain. A B D C 868 DAB > ?

REASONING Use the photo to copy and complete the statement. Explain. A B D C 868 BCA > ?

REASONING Use the photo to copy and complete the statement. Explain. A B D C 868 m ABC 5 ?

REASONING Use the photo to copy and complete the statement. Explain. A B D C 868 m CAB 5 ?

REASONING Use the photo to copy and complete the statement. Explain. A B D C 868 m CAD 5 ?

USING A DIAGRAM Find the indicated measure in ~EFGH. Explain. E F 458 858 60 m EJF

USING A DIAGRAM Find the indicated measure in ~EFGH. Explain. E F 458 858 60 m EGF

USING A DIAGRAM Find the indicated measure in ~EFGH. Explain. E F 458 858 60 m HFG

USING A DIAGRAM Find the indicated measure in ~EFGH. Explain. E F 458 858 60 m GEF

USING A DIAGRAM Find the indicated measure in ~EFGH. Explain. E F 458 858 60 m HGF

USING A DIAGRAM Find the indicated measure in ~EFGH. Explain. E F 458 858 60 m EHG

MULTIPLE CHOICE In parallelogram ABCD, AB 5 14 inches and BC 5 20 inches. What is the perimeter (in inches) of ~ABCD? A 28 B 40 C 68 D 280

ALGEBRA The measure of one interior angle of a parallelogram is 0.25 times the measure of another angle. Find the measure of each angle.

ALGEBRA The measure of one interior angle of a parallelogram is 50 degrees more than 4 times the measure of another angle. Find the measure of each angle.

ERROR ANALYSIS In ~ABCD, m B 5 508. A student says that m A 5 508. Explain why this statement is incorrect.

USING A DIAGRAM In the diagram, QRST and STUV are parallelograms. Find the values of x and y. Explain your reasoning. P 20 R T S 408 y8 8

FINDING A PERIMETER The sides of ~MNPQ are represented by the expressions below. Sketch ~MNPQ and find its perimeter. MQ 5 22x 1 37 QP 5 y 1 14 NP 5 x 2 5 MN 5 4y 1 5

SHORT RESPONSE In ABCD, m B 5 1248, m A 5 668, and m C 5 1248. Explain why ABCD cannot be a parallelogram.

FINDING ANGLE MEASURES In ~LMNP shown at the right, m MLN 5 328, m NLP 5 (x2 )8, m MNP 5 12x8, and MNP is an acute angle. Find m NLP. P M

CHALLENGE Points A(1, 2), B(3, 6), and C(6, 4) are three vertices of ~ABCD. Find the coordinates of each point that could be vertex D. Sketch each possible parallelogram in a separate coordinate plane. Justify your answers.

AIRPLANE The diagram shows the mechanism for opening the canopy on a small airplane. Two pivot arms attach at four pivot points A, B, C, and D. These points form the vertices of a parallelogram. Find m D when m C 5 408. Explain your reasoning.

MIRROR The mirror shown is attached to the wall by an arm that can extend away from the wall. In the figure, points P, Q, R, and S are the vertices of a parallelogram. This parallelogram is one of several that change shape as the mirror is extended. a. If PQ 5 3 inches, find RS. b. If m Q 5 708, what is m S? c. What happens to m P as m Q increases? What happens to QS as m Q decreases? Explain.

USING RATIOS In ~LMNO, the ratio of LM to MN is 4 : 3. Find LM if the perimeter of LMNO is 28.

OPEN-ENDED MATH Draw a triangle. Copy the triangle and combine the two triangles to form a quadrilateral. Show that the quadrilateral is a parallelogram. Then show how you can make additional copies of the triangle to form a larger parallelogram that is similar to the first parallelogram. Justify your method.

PROVING THEOREM 8.4 Use the diagram of quadrilateral ABCD with the auxiliary line segment drawn to write a two-column proof of Theorem 8.4. GIVEN c ABCD is a parallelogram. PROVE c A > C, B > D B C A D

PROVING THEOREM 8.5 Use properties of parallel lines to prove Theorem 8.5. GIVEN c PQRS is a parallelogram. PROVE c x8 1 y8 5 1808 P P S R x8 y8 y8 x8

PROVING THEOREM 8.6 Theorem 8.6 states that if a quadrilateral is a parallelogram, then its diagonals bisect each other. Write a two-column proof of Theorem 8.6.

CHALLENGE Suppose you choose a point on the base of an isosceles triangle. You draw segments from that point perpendicular to the legs of the triangle. Prove that the sum of the lengths of those segments is equal to the length of the altitude drawn to one leg. GIVEN c nABC is isosceles with base }AC , }AF is the altitude drawn to }BC , }DE }AB , }DG }BC PROVE c For D anywhere on }AC , DE 1 DG 5 AF. G F B A D E

Tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. (p. 171) Line 1: (2, 4), (4, 1) Line 2: (5, 7), (9, 0)

Tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. (p. 171) Line 1: (26, 7), (22, 3) Line 2: (9, 21), (2, 6)

Tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. (p. 171) Line 1: (23, 0), (26, 5) Line 2: (3, 25), (5, 210)

Decide if the side lengths form a triangle. If so, would the triangle be acute, right, or obtuse? (p. 441) 9, 13, and 6

Decide if the side lengths form a triangle. If so, would the triangle be acute, right, or obtuse? (p. 441) 10, 12, and 7

Decide if the side lengths form a triangle. If so, would the triangle be acute, right, or obtuse? (p. 441) 5, 9, and } 106

Decide if the side lengths form a triangle. If so, would the triangle be acute, right, or obtuse? (p. 441) 8, 12, and 4

Decide if the side lengths form a triangle. If so, would the triangle be acute, right, or obtuse? (p. 441) 24, 10, and 26

Decide if the side lengths form a triangle. If so, would the triangle be acute, right, or obtuse? (p. 441) 9, 10, and 11

Find the value of x. Write your answer in simplest radical form. (p. 457) 12 608 x

Find the value of x. Write your answer in simplest radical form. (p. 457) 15 3

Find the value of x. Write your answer in simplest radical form. (p. 457) 8 458 x

VOCABULARY Explain how knowing that }AB i }CD and }AD i }BC allows you to show that quadrilateral ABCD is a parallelogram.

WRITING A quadrilateral has four congruent sides. Is the quadrilateral a parallelogram? Justify your answer.

ERROR ANALYSIS A student claims that because two pairs of sides are congruent, quadrilateral DEFG shown at the right is a parallelogram. Describe the error that the student is making. DEFG is a parallelogram. E D 5 5 7

REASONING What theorem can you use to show that the quadrilateral is a parallelogram? 1008 100

REASONING What theorem can you use to show that the quadrilateral is a parallelogram? 20 14 14

REASONING What theorem can you use to show that the quadrilateral is a parallelogram?

SHORT RESPONSE When you shift gears on a bicycle, a mechanism called a derailleur moves the chain to a new gear. For the derailleur shown below, JK 5 5.5 cm, KL 5 2 cm, ML 5 5.5 cm, and MJ 5 2 cm. Explain why }JK and}ML are always parallel as the derailleur moves.

ALGEBRA For what value of x is the quadrilateral a parallelogram? 2x 1 3 x 1 7

ALGEBRA For what value of x is the quadrilateral a parallelogram? 4x 1 2 5x 2 6

ALGEBRA For what value of x is the quadrilateral a parallelogram? 6x 3x 1 2

COORDINATE GEOMETRY The vertices of quadrilateral ABCD are given. Draw ABCD in a coordinate plane and show that it is a parallelogram. A(0, 1), B(4, 4), C(12, 4), D(8, 1)

COORDINATE GEOMETRY The vertices of quadrilateral ABCD are given. Draw ABCD in a coordinate plane and show that it is a parallelogram. A(23, 0), B(23, 4), C(3, 21), D(3, 25)

COORDINATE GEOMETRY The vertices of quadrilateral ABCD are given. Draw ABCD in a coordinate plane and show that it is a parallelogram. A(22, 3), B(25, 7), C(3, 6), D(6, 2)

COORDINATE GEOMETRY The vertices of quadrilateral ABCD are given. Draw ABCD in a coordinate plane and show that it is a parallelogram. A(25, 0), B(0, 4), C(3, 0), D(22, 24)

REASONING Describe how to prove that ABCD is a parallelogram. B D C

REASONING Describe how to prove that ABCD is a parallelogram. A B D C

REASONING Describe how to prove that ABCD is a parallelogram. A B D C

MULTIPLE CHOICE In quadrilateral WXYZ,}WZ and }XY are congruent and parallel. Which statement below is not necessarily true? A m Y 1 m W 5 1808 B X > Z C }WX > }ZY D }WX i }ZY

ALGEBRA For what value of x is the quadrilateral a parallelogram? 668 668

ALGEBRA For what value of x is the quadrilateral a parallelogram? x8 3x8

ALGEBRA For what value of x is the quadrilateral a parallelogram? x 1 10)8 (2x 1 20)8

BICONDITIONALS Write the indicated theorems as a biconditional statement. Theorem 8.3, page 515 and Theorem 8.7, page 522

BICONDITIONALS Write the indicated theorems as a biconditional statement. Theorem 8.4, page 515 and Theorem 8.8, page 522

REASONING Follow the steps below to draw a parallelogram. Explain why this method works. State a theorem to support your answer. STEP 1 Use a ruler to draw two segments STEP 2 Connect the endpoints of the that intersect at their midpoints. segments to form a quadrilateral.

COORDINATE GEOMETRY Three of the vertices of ~ABCD are given. Find the coordinates of point D. Show your method. A(22, 23), B(4, 23), C(3, 2), D(x, y)

COORDINATE GEOMETRY Three of the vertices of ~ABCD are given. Find the coordinates of point D. Show your method. . A(24, 1), B(21, 5), C(6, 5), D(x, y)

COORDINATE GEOMETRY Three of the vertices of ~ABCD are given. Find the coordinates of point D. Show your method. A(24, 4), B(4, 6), C(3, 21), D(x, y)

COORDINATE GEOMETRY Three of the vertices of ~ABCD are given. Find the coordinates of point D. Show your method. A(21, 0), B(0, 24), C(8, 26), D(x, y)

CONSTRUCTION There is more than one way to use a compass and a straightedge to construct a parallelogram. Describe a method that uses Theorem 8.7 or Theorem 8.9. Then use your method to construct a parallelogram.

CHALLENGE In the diagram, ABCD is a parallelogram, BF 5 DE 5 12, and CF 5 8. Find AE. Explain your reasoning. A B C F E D

AUTOMOBILE REPAIR The diagram shows an automobile lift. A bus drives on to the ramp (}EG ). Levers (}EK , }FJ , and}GH ) raise the bus. In the diagram, EG}> }KH and EK 5 FJ 5 GH. Also, F is the midpoint of }EG , and J is the midpoint of }KH . a. Identify all the quadrilaterals in the automobile lift. Explain how you know that each one is a parallelogram. b. Explain why }EG is always parallel to }KH .

MUSIC STAND A music stand can be folded up, as shown below. In the diagrams, A > EFD, D > AEF, C > BEF, and B > CFE. Explain why }AD and }BC remain parallel as the stand is folded up. Which other labeled segments remain parallel? A B D C E F A D B C E F

PROVING THEOREM 8.9 Use the diagram of PQRS with the auxiliary line segment drawn. Copy and complete the flow proof of Theorem 8.9. GIVEN c }QR i }PS , }QR > }PS PROVE c PQRS is a parallelogram. P R P S }QR i }PS Given PSQ > RQS ? QS}> }QS ? nRSQ > nPQS ? PQRS is a ~. ? ? }QR > }PS ? }RS > }PQ

REASONING A student claims incorrectly that the marked information can be used to show that the figure is a parallelogram. Draw a quadrilateral with the marked properties that is clearly not a parallelogram. Explain.

REASONING A student claims incorrectly that the marked information can be used to show that the figure is a parallelogram. Draw a quadrilateral with the marked properties that is clearly not a parallelogram. Explain.

REASONING A student claims incorrectly that the marked information can be used to show that the figure is a parallelogram. Draw a quadrilateral with the marked properties that is clearly not a parallelogram. Explain. 8 8

EXTENDED RESPONSE Theorem 8.5 states that if a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Write the converse of Theorem 8.5. Then write a plan for proving the converse of Theorem 8.5. Include a diagram.

PROVING THEOREM 8.8 Prove Theorem 8.8. GIVEN c A > C, B > D PROVE c ABCD is a parallelogram. Hint: Let x8 represent m A and m C, and let y8 represent m B and m D. Write and simplify an equation involving x and y. A D C

PROVING THEOREM 8.10 Prove Theorem 8.10. GIVEN c Diagonals }JL and}KM bisect each other. PROVE c JKLM is a parallelogram. K L J M P

PROOF Use the diagram at the right. GIVEN c DEBF is a parallelogram, AE 5 CF PROVE c ABCD is a parallelogram. A E B D

REASONING In the diagram, the midpoints of the sides of a quadrilateral have been joined to form what appears to be a parallelogram. Show that a quadrilateral formed by connecting the midpoints of the sides of any quadrilateral is always a parallelogram. (Hint: Draw a diagram. Include a diagonal of the larger quadrilateral. Show how two sides of the smaller quadrilateral are related to the diagonal.)

CHALLENGE Show that if ABCD is a parallelogram with its diagonals intersecting at E, then you can connect the midpoints F, G, H, and J of }AE , }BE , }CE , and }DE , respectively, to form another parallelogram, FGHJ. A B E D C F G J H

In Exercises 4345, draw a figure that fits the description. (p. 42) A quadrilateral that is equilateral but not equiangular

In Exercises 4345, draw a figure that fits the description. (p. 42) A quadrilateral that is equiangular but not equilateral

In Exercises 4345, draw a figure that fits the description. (p. 42) A quadrilateral that is concave

The width of a rectangle is 4 centimeters less than its length. The perimeter of the rectangle is 42 centimeters. Find its area. (p. 49)

Find the values of x and y in the triangle shown at the right. Write your answers in simplest radical form. (p. 457) 308 x 4

VOCABULARY What is another name for an equilateral rectangle?

WRITING Do you have enough information to identify the figure at the right as a rhombus? Explain. W X Z Y

RHOMBUSES For any rhombus JKLM, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. L > M

RHOMBUSES For any rhombus JKLM, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. K > M

RHOMBUSES For any rhombus JKLM, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. JK > }KL

RHOMBUSES For any rhombus JKLM, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. JM > }KL

RHOMBUSES For any rhombus JKLM, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. JL >}KM

RHOMBUSES For any rhombus JKLM, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. JKM > LKM

RECTANGLES For any rectangle WXYZ, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. W > X

RECTANGLES For any rectangle WXYZ, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. WX > }YZ

RECTANGLES For any rectangle WXYZ, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. WX}> }XY

RECTANGLES For any rectangle WXYZ, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. WY > }XZ

RECTANGLES For any rectangle WXYZ, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. WY }XZ

RECTANGLES For any rectangle WXYZ, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. WXZ > YXZ

CLASSIFYING Classify the quadrilateral. Explain your reasoning.

CLASSIFYING Classify the quadrilateral. Explain your reasoning.

CLASSIFYING Classify the quadrilateral. Explain your reasoning. 1408 408

USING PROPERTIES Sketch rhombus STUV. Describe everything you know about the rhombus.

USING PROPERTIES Name each quadrilateralparallelogram, rectangle, rhombus, and squarefor which the statement is true. It is equiangular.

USING PROPERTIES Name each quadrilateralparallelogram, rectangle, rhombus, and squarefor which the statement is true. It is equiangular and equilateral.

USING PROPERTIES Name each quadrilateralparallelogram, rectangle, rhombus, and squarefor which the statement is true. Its diagonals are perpendicular.

USING PROPERTIES Name each quadrilateralparallelogram, rectangle, rhombus, and squarefor which the statement is true. Opposite sides are congruent.

USING PROPERTIES Name each quadrilateralparallelogram, rectangle, rhombus, and squarefor which the statement is true. The diagonals bisect each other.

USING PROPERTIES Name each quadrilateralparallelogram, rectangle, rhombus, and squarefor which the statement is true. The diagonals bisect opposite angles.

ERROR ANALYSIS Quadrilateral PQRS is a rectangle. Describe and correct the error made in finding the value of x. 7x 2 4 5 3x 1 14 4x 5 18 x 5 4.5

ALGEBRA Classify the special quadrilateral. Explain your reasoning. Then find the values of x and y. C 3y x8 1048 y

ALGEBRA Classify the special quadrilateral. Explain your reasoning. Then find the values of x and y. M 5x 2 9 2y 1 35 K

ALGEBRA Classify the special quadrilateral. Explain your reasoning. Then find the values of x and y. 5x8 2y 10 P S P R (3x 1 18)

ALGEBRA Classify the special quadrilateral. Explain your reasoning. Then find the values of x and y. 2y 1 1 (4x 1 7)8 (5x 2 6)8 y 1 3 E

SHORT RESPONSE The diagonals of a rhombus are 6 inches and 8 inches. What is the perimeter of the rhombus? Explain.

MULTIPLE CHOICE Rectangle ABCD is similar to rectangle FGHJ. If AC 5 5, CD 5 4, and FM 5 5, what is HJ? A 4 B 5 C 8 D 10 J H F G C E M D A

RHOMBUS The diagonals of rhombus ABCD intersect at E. Given that m BAC 5 538 and DE 5 8, find the indicated measure. A B 538 8 m DAC

RHOMBUS The diagonals of rhombus ABCD intersect at E. Given that m BAC 5 538 and DE 5 8, find the indicated measure. A B 538 8 m AED

RHOMBUS The diagonals of rhombus ABCD intersect at E. Given that m BAC 5 538 and DE 5 8, find the indicated measure. A B 538 8 m ADC

RHOMBUS The diagonals of rhombus ABCD intersect at E. Given that m BAC 5 538 and DE 5 8, find the indicated measure. A B 538 8 DB

RHOMBUS The diagonals of rhombus ABCD intersect at E. Given that m BAC 5 538 and DE 5 8, find the indicated measure. A B 538 8 AE

RHOMBUS The diagonals of rhombus ABCD intersect at E. Given that m BAC 5 538 and DE 5 8, find the indicated measure. A B 538 8 AC

RECTANGLE The diagonals of rectangle QRST intersect at P. Given that m PTS 5 348 and QS 5 10, find the indicated measure. P R 348 m SRT

RECTANGLE The diagonals of rectangle QRST intersect at P. Given that m PTS 5 348 and QS 5 10, find the indicated measure. P R 348 m QPR

RECTANGLE The diagonals of rectangle QRST intersect at P. Given that m PTS 5 348 and QS 5 10, find the indicated measure. P R 348 QP

RECTANGLE The diagonals of rectangle QRST intersect at P. Given that m PTS 5 348 and QS 5 10, find the indicated measure. P R 348 RP

RECTANGLE The diagonals of rectangle QRST intersect at P. Given that m PTS 5 348 and QS 5 10, find the indicated measure. P R 348 QR

RECTANGLE The diagonals of rectangle QRST intersect at P. Given that m PTS 5 348 and QS 5 10, find the indicated measure. P R 348 RS

SQUARE The diagonals of square LMNP intersect at K. Given that LK 5 1, find the indicated measure. N K P L m MKN

SQUARE The diagonals of square LMNP intersect at K. Given that LK 5 1, find the indicated measure. N K P L m LMK

SQUARE The diagonals of square LMNP intersect at K. Given that LK 5 1, find the indicated measure. N K P L m LPK

SQUARE The diagonals of square LMNP intersect at K. Given that LK 5 1, find the indicated measure. N K P L KN

SQUARE The diagonals of square LMNP intersect at K. Given that LK 5 1, find the indicated measure. N K P L MP

SQUARE The diagonals of square LMNP intersect at K. Given that LK 5 1, find the indicated measure. N K P L LP

COORDINATE GEOMETRY Use the given vertices to graph ~JKLM. Classify ~JKLM and explain your reasoning. Then find the perimeter of ~JKLM. J(24, 2), K(0, 3), L(1, 21), M(23, 22)

COORDINATE GEOMETRY Use the given vertices to graph ~JKLM. Classify ~JKLM and explain your reasoning. Then find the perimeter of ~JKLM. J(22, 7), K(7, 2), L(22, 23), M(211, 2)

REASONING Are all rhombuses similar? Are all squares similar? Explain your reasoning.

CHALLENGE Quadrilateral ABCD shown at the right is a rhombus. Given that AC 5 10 and BD 5 16, find all side lengths and angle measures. Explain your reasoning. C E D A

MULTI-STEP PROBLEM In the window shown at the right, }BD > }DF > }BH > }HF . Also, HAB, BCD, DEF, and FGH are right angles. a. Classify HBDF and ACEG. Explain your reasoning. b. What can you conclude about the lengths of the diagonals }AE and GC} ? Given that these diagonals intersect at J, what can you conclude about the lengths of AJ } , }JE , }CJ , and }JG ? Explain.

PATIO You want to mark off a square region in your yard for a patio. You use a tape measure to mark off a quadrilateral on the ground. Each side of the quadrilateral is 2.5 meters long. Explain how you can use the tape measure to make sure that the quadrilateral you drew is a square.

PROVING THEOREM 8.11 Use the plan for proof below to write a paragraph proof for the converse statement of Theorem 8.11. GIVEN c ABCD is a rhombus. PROVE c }AC }BD Plan for Proof Because ABCD is a parallelogram, its diagonals bisect each other at X. Show that nAXB > nCXB. Then show that }AC and }BD intersect to form congruent adjacent angles, AXB and CXB. X D C A

PROVING COROLLARIES Write the corollary as a conditional statement and its converse. Then explain why each statement is true. Rhombus Corollary

PROVING COROLLARIES Write the corollary as a conditional statement and its converse. Then explain why each statement is true. Rectangle Corollary

PROVING COROLLARIES Write the corollary as a conditional statement and its converse. Then explain why each statement is true. Square Corollary

PROVING THEOREM 8.12 In Exercises 60 and 61, prove both parts of Theorem 8.12. GIVEN c PQRS is a parallelogram. PR} bisects SPQ and QRS. } bisects PSR and RQP PROVE c PQRS is a rhombus. P R T P S

PROVING THEOREM 8.12 In Exercises 60 and 61, prove both parts of Theorem 8.12. GIVEN c WXYZ is a rhombus. PROVE c }WY bisects ZWX and XYZ. ZX}bisects WZY and YXW. Z Y V W

EXTENDED RESPONSE In ABCD, }AB i }CD , and }DB bisects ADC. a. Show that ABD > CDB. What can you conclude about ADB and ABD? What can you conclude about }AB and }AD ? Explain. b. Suppose you also know that }AD > }BC . Classify ABCD. Explain. D C A

PROVING THEOREM 8.13 Write a coordinate proof of the following statement, which is part of Theorem 8.13. If a quadrilateral is a rectangle, then its diagonals are congruent.

CHALLENGE Write a coordinate proof of part of Theorem 8.13. GIVEN c DFGH is a parallelogram,}DG > }HF PROVE c DFGH is a rectangle. Plan for Proof Write the coordinates of the vertices in terms of a and b. Find and compare the slopes of the sides. y O D(a, ?) F(b, 0) H(?, ?) G(?, ?)

In nJKL, KL 5 54.2 centimeters. Point M is the midpoint of }JK and N is the midpoint of }JL . Find MN. (p. 295)

Find the sine and cosine of the indicated angle. Write each answer as a fraction and a decimal. (p. 473) 50 T S R

Find the sine and cosine of the indicated angle. Write each answer as a fraction and a decimal. (p. 473) 50 T S T

Find the value of x. (p. 507) 288 1068 1278 x

Find the value of x. (p. 507) 1208 1638 1558 1178 1088

Find the value of x. (p. 507) 758 628 888

VOCABULARY In trapezoid PQRS, }PQ i }RS . Sketch PQRS and identify its bases and its legs.

WRITING Describe the differences between a kite and a trapezoid.

COORDINATE PLANE Points A, B, C, and D are the vertices of a quadrilateral. Determine whether ABCD is a trapezoid. A(0, 4), B(4, 4), C(8, 22), D(2, 1)

COORDINATE PLANE Points A, B, C, and D are the vertices of a quadrilateral. Determine whether ABCD is a trapezoid. A(25, 0), B(2, 3), C(3, 1), D(22, 22)

COORDINATE PLANE Points A, B, C, and D are the vertices of a quadrilateral. Determine whether ABCD is a trapezoid. A(2, 1), B(6, 1), C(3, 23), D(21, 24)

COORDINATE PLANE Points A, B, C, and D are the vertices of a quadrilateral. Determine whether ABCD is a trapezoid. A(23, 3), B(21, 1), C(1, 24), D(23, 0)

FINDING ANGLE MEASURES Find m J, m L, and m M. K L 508

FINDING ANGLE MEASURES Find m J, m L, and m M. J L K 1008

FINDING ANGLE MEASURES Find m J, m L, and m M. K 1188

REASONING Determine whether the quadrilateral is a trapezoid. Explain. A B D C

REASONING Determine whether the quadrilateral is a trapezoid. Explain. H G E

REASONING Determine whether the quadrilateral is a trapezoid. Explain. M L J

FINDING MIDSEGMENTS Find the length of the midsegment of the trapezoid. 18 10 M N

FINDING MIDSEGMENTS Find the length of the midsegment of the trapezoid. 21 25 M N

FINDING MIDSEGMENTS Find the length of the midsegment of the trapezoid. 57 76 N M

MULTIPLE CHOICE Which statement is not always true? A The base angles of an isosceles trapezoid are congruent. B The midsegment of a trapezoid is parallel to the bases. C The bases of a trapezoid are parallel. D The legs of a trapezoid are congruent.

ERROR ANALYSIS Describe and correct the error made in finding m A. B D A 120 50 Opposite angles of a kite are congruent, so m A 5 508.

ANGLES OF KITES EFGH is a kite. Find m G. E G H 1008 408

ANGLES OF KITES EFGH is a kite. Find m G. E 608 1108

ANGLES OF KITES EFGH is a kite. Find m G. H E 1508

DIAGONALS OF KITES Use Theorem 8.18 and the Pythagorean Theorem to find the side lengths of the kite. Write the lengths in simplest radical form. 3 3 5 3 X

DIAGONALS OF KITES Use Theorem 8.18 and the Pythagorean Theorem to find the side lengths of the kite. Write the lengths in simplest radical form. 12 6 6 4 X

DIAGONALS OF KITES Use Theorem 8.18 and the Pythagorean Theorem to find the side lengths of the kite. Write the lengths in simplest radical form. 9 10 10 5 X W

ERROR ANALYSIS In trapezoid ABCD, MN}is the midsegment. Describe and correct the error made in finding DC. DC 5 AB 2 MN DC 5 14 2 8 DC 5 6 14 8 D C M N A

ALGEBRA Find the value of x. 10 7 2x

ALGEBRA Find the value of x. 3x 1 1 12.5 15

ALGEBRA Find the value of x. 5x 18.7 12x 2 1.7

SHORT RESPONSE The points M(23, 5), N(21, 5), P(3, 21), and Q(25, 21) form the vertices of a trapezoid. Draw MNPQ and find MP and NQ. What do your results tell you about the trapezoid? Explain.

DRAWING In trapezoid JKLM, }JK i}LM and JK 5 17. The midsegment of JKLM is }XY , and XY 5 37. Sketch JKLM and its midsegment. Then find LM.

RATIOS The ratio of the lengths of the bases of a trapezoid is 1 : 3. The length of the midsegment is 24. Find the lengths of the bases.

MULTIPLE CHOICE In trapezoid PQRS, }PQ i }RS and}MN is the midsegment of PQRS. If RS 5 5 p PQ, what is the ratio of MN to RS? A 3:5 B 5:3 C 2:1 D 3:1

CHALLENGE The figure shown at the right is a trapezoid with its midsegment. Find all the possible values of x. What is the length of the midsegment? Explain. (The figure may not be drawn to scale.) 36 x2 7x 2 6

REASONING Explain why a kite and a general quadrilateral are the only quadrilaterals that can be concave.

FURNITURE In the photograph of a chest of drawers,}HC is the midsegment of trapezoid ABDG,}GD is the midsegment of trapezoid HCEF, AB 5 13.9 centimeters, and GD 5 50.5 centimeters. Find HC. Then find FE. A B H C G D F E

GRAPHIC DESIGN You design a logo in the shape of a convex kite. The measure of one angle of the kite is 908. The measure of another angle is 308. Sketch a kite that matches this description. Give the measures of all the angles and mark any congruent sides.

EXTENDED RESPONSE The bridge below is designed to fold up into an octagon shape. The diagram shows a section of the bridge. a. Classify the quadrilaterals shown in the diagram. b. As the bridge folds up, what happens to the length of }BF ? What happens to m BAF, m ABC, m BCF, and m CFA? c. Given m CFE 5 658, find m DEF, m FCD, and m CDE. Explain.

PROVING THEOREM 8.14 Use the diagram and the auxiliary segment to prove Theorem 8.14. In the diagram, }EC is drawn parallel to }AB . GIVEN c ABCD is an isosceles trapezoid, }BC i }AD PROVE c A > D, B > BCD Hint: Find a way to show that nECD is an isosceles triangle. A E D B

PROVING THEOREM 8.15 Use the diagram and the auxiliary segment to prove Theorem 8.15. In the diagram, }JG is drawn parallel to }EF . GIVEN c EFGH is a trapezoid, }FG i }EH , E > H PROVE c EFGH is an isosceles trapezoid. Hint: Find a way to show that nJGH is an isosceles triangle. E J H F

PROVING THEOREM 8.16 Prove part of Theorem 8.16. GIVEN c JKLM is an isosceles trapezoid. }KL i }JM , }JK >}LM PROVE c }JL >}KM J M K

REASONING In the diagram below, }BG is the midsegment of nACD and }GE is the midsegment of nADF. Explain why the midsegment of trapezoid ACDF is parallel to each base and why its length is one half the sum of the lengths of the bases. D F B G G A A E C D D

PROVING THEOREM 8.18 Prove Theorem 8.18. GIVEN c ABCD is a kite. }AB > }CB , }AD > }CD PROVE c }AC }BD D E C A B

PROVING THEOREM 8.19 Write a paragraph proof of Theorem 8.19. GIVEN c EFGH is a kite. }EF > }GF , }EH >}GH PROVE c E > G, F H Plan for Proof First show that E > G. Then use an indirect argument to show that F H: If F > H, then EFGH is a parallelogram. But opposite sides of a parallelogram are congruent. This result contradicts the definition of a kite. H G E F

CHALLENGE In Exercise 39, you proved that part of Theorem 8.16 is true. Write the other part of Theorem 8.16 as a conditional statement. Then prove that the statement is true.

Place a right triangle in a coordinate plane in a way that is convenient for finding side lengths. Assign coordinates to each vertex. (p. 295)

Use the diagram to complete the proportion. (p. 449) B C D A AB AC 5 } ? AB

Use the diagram to complete the proportion. (p. 449) B C D A AB BC 5 } BD ?

Three of the vertices of ~ABCD are given. Find the coordinates of point D. Show your method. (p. 522) A(21, 22), B(4, 22), C(6, 2), D(x, y)

Three of the vertices of ~ABCD are given. Find the coordinates of point D. Show your method. (p. 522) A(1, 4), B(0, 1), C(4, 1), D(x, y)

VOCABULARY Copy and complete: A quadrilateral that has exactly one pair of parallel sides and diagonals that are congruent is a(n) ? .

WRITING Describe three methods you could use to prove that a parallelogram is a rhombus.

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box if the shape always has the given property.Property ~ Rectangle Rhombus Square Kite Trapezoid All sides are >. ? ? ? ?? ? Both pairs of opp. sides are >.? ? ? ? ? ? Both pairs of opp. sides are i.? ? ? ? ? ? Exactly 1 pair of opp. sides are i. ? ? ? ?? ? All ? are >. ? ? ? ?? ? Exactly 1 pair of opp. ? are >.? ? ? ? ? ? Diagonals are . ? ? ? ?? ? Diagonals are >. ? ? ? ?? ? Diagonals bisect each other. ? ? ? ? ? ?

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box if the shape always has the given property. Property ~ Rectangle Rhombus Square Kite Trapezoid All sides are >. ? ? ? ?? ? Both pairs of opp. sides are >.? ? ? ? ? ? Both pairs of opp. sides are i.? ? ? ? ? ? Exactly 1 pair of opp. sides are i. ? ? ? ?? ? All ? are >. ? ? ? ?? ? Exactly 1 pair of opp. ? are >.? ? ? ? ? ? Diagonals are . ? ? ? ?? ? Diagonals are >. ? ? ? ?? ? Diagonals bisect each other. ? ? ? ? ? ?

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box if the shape always has the given property. Property ~ Rectangle Rhombus Square Kite Trapezoid All sides are >. ? ? ? ?? ? Both pairs of opp. sides are >.? ? ? ? ? ? Both pairs of opp. sides are i.? ? ? ? ? ? Exactly 1 pair of opp. sides are i. ? ? ? ?? ? All ? are >. ? ? ? ?? ? Exactly 1 pair of opp. ? are >.? ? ? ? ? ? Diagonals are . ? ? ? ?? ? Diagonals are >. ? ? ? ?? ? Diagonals bisect each other. ? ? ? ? ? ?

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box if the shape always has the given property. Property ~ Rectangle Rhombus Square Kite Trapezoid All sides are >. ? ? ? ?? ? Both pairs of opp. sides are >.? ? ? ? ? ? Both pairs of opp. sides are i.? ? ? ? ? ? Exactly 1 pair of opp. sides are i. ? ? ? ?? ? All ? are >. ? ? ? ?? ? Exactly 1 pair of opp. ? are >.? ? ? ? ? ? Diagonals are . ? ? ? ?? ? Diagonals are >. ? ? ? ?? ? Diagonals bisect each other. ? ? ? ? ? ?

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box if the shape always has the given property. Property ~ Rectangle Rhombus Square Kite Trapezoid All sides are >. ? ? ? ?? ? Both pairs of opp. sides are >.? ? ? ? ? ? Both pairs of opp. sides are i.? ? ? ? ? ? Exactly 1 pair of opp. sides are i. ? ? ? ?? ? All ? are >. ? ? ? ?? ? Exactly 1 pair of opp. ? are >.? ? ? ? ? ? Diagonals are . ? ? ? ?? ? Diagonals are >. ? ? ? ?? ? Diagonals bisect each other. ? ? ? ? ? ?

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box if the shape always has the given property. Property ~ Rectangle Rhombus Square Kite Trapezoid All sides are >. ? ? ? ?? ? Both pairs of opp. sides are >.? ? ? ? ? ? Both pairs of opp. sides are i.? ? ? ? ? ? Exactly 1 pair of opp. sides are i. ? ? ? ?? ? All ? are >. ? ? ? ?? ? Exactly 1 pair of opp. ? are >.? ? ? ? ? ? Diagonals are . ? ? ? ?? ? Diagonals are >. ? ? ? ?? ? Diagonals bisect each other. ? ? ? ? ? ?

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box if the shape always has the given property. Property ~ Rectangle Rhombus Square Kite Trapezoid All sides are >. ? ? ? ?? ? Both pairs of opp. sides are >.? ? ? ? ? ? Both pairs of opp. sides are i.? ? ? ? ? ? Exactly 1 pair of opp. sides are i. ? ? ? ?? ? All ? are >. ? ? ? ?? ? Exactly 1 pair of opp. ? are >.? ? ? ? ? ? Diagonals are . ? ? ? ?? ? Diagonals are >. ? ? ? ?? ? Diagonals bisect each other. ? ? ? ? ? ?

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box if the shape always has the given property. Property ~ Rectangle Rhombus Square Kite Trapezoid All sides are >. ? ? ? ?? ? Both pairs of opp. sides are >.? ? ? ? ? ? Both pairs of opp. sides are i.? ? ? ? ? ? Exactly 1 pair of opp. sides are i. ? ? ? ?? ? All ? are >. ? ? ? ?? ? Exactly 1 pair of opp. ? are >.? ? ? ? ? ? Diagonals are . ? ? ? ?? ? Diagonals are >. ? ? ? ?? ? Diagonals bisect each other. ? ? ? ? ? ?

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box if the shape always has the given property. Property ~ Rectangle Rhombus Square Kite Trapezoid All sides are >. ? ? ? ?? ? Both pairs of opp. sides are >.? ? ? ? ? ? Both pairs of opp. sides are i.? ? ? ? ? ? Exactly 1 pair of opp. sides are i. ? ? ? ?? ? All ? are >. ? ? ? ?? ? Exactly 1 pair of opp. ? are >.? ? ? ? ? ? Diagonals are . ? ? ? ?? ? Diagonals are >. ? ? ? ?? ? Diagonals bisect each other. ? ? ? ? ? ?

ERROR ANALYSIS Describe and correct the error in classifying the quadrilateral. A B D C 59 125 55 B and C are supplements, so }AB i }CD . So, ABCD is a parallelogram.

MULTIPLE CHOICE What is the most specific name for the quadrilateral shown at the right? A Rectangle B Parallelogram C Trapezoid D Isosceles trapezoid

CLASSIFYING QUADRILATERALS Give the most specific name for the quadrilateral. Explain. J M K L

CLASSIFYING QUADRILATERALS Give the most specific name for the quadrilateral. Explain. P P S R

CLASSIFYING QUADRILATERALS Give the most specific name for the quadrilateral. Explain. W Y Z

DRAWING Draw a quadrilateral with congruent diagonals and exactly one pair of congruent sides. What is the most specific name for this quadrilateral?

IDENTIFYING QUADRILATERALS Tell whether enough information is given in the diagram to classify the quadrilateral by the indicated name. Explain. Rhombus A D B

IDENTIFYING QUADRILATERALS Tell whether enough information is given in the diagram to classify the quadrilateral by the indicated name. Explain. Isosceles trapezoid E F 1118 708 708

IDENTIFYING QUADRILATERALS Tell whether enough information is given in the diagram to classify the quadrilateral by the indicated name. Explain. Square M L J

COORDINATE PLANE Points P, Q, R, and S are the vertices of a quadrilateral. Give the most specific name for PQRS. Justify your answer. P(1, 0), Q(1, 2), R(6, 5), S(3, 0)

COORDINATE PLANE Points P, Q, R, and S are the vertices of a quadrilateral. Give the most specific name for PQRS. Justify your answer. P(2, 1), Q(6, 1), R(5, 8), S(3, 8)

COORDINATE PLANE Points P, Q, R, and S are the vertices of a quadrilateral. Give the most specific name for PQRS. Justify your answer. P(2, 7), Q(6, 9), R(9, 3), S(5, 1)

COORDINATE PLANE Points P, Q, R, and S are the vertices of a quadrilateral. Give the most specific name for PQRS. Justify your answer. P(1, 7), Q(5, 8), R(6, 2), S(2, 1)

TECHNOLOGY Use geometry drawing software to draw points A, B, C, and segments AC and BC. Draw a circle with center A and radius AC. Draw a circle with center B and radius BC. Label the other intersection of the circles D. Draw }BD and AD} . a. Drag point A, B, C, or D to change the shape of ABCD. What types of quadrilaterals can be formed? b. Are there types of quadrilaterals that cannot be formed? Explain. C D A B

DEVELOPING PROOF Which pairs of segments or angles must be congruent so that you can prove that ABCD is the indicated quadrilateral? Explain. There may be more than one right answer. Square D C A

DEVELOPING PROOF Which pairs of segments or angles must be congruent so that you can prove that ABCD is the indicated quadrilateral? Explain. There may be more than one right answer. Isosceles trapezoid A B 608 1208

DEVELOPING PROOF Which pairs of segments or angles must be congruent so that you can prove that ABCD is the indicated quadrilateral? Explain. There may be more than one right answer. Parallelogram D A B

TRAPEZOIDS In Exercises 2931, determine whether there is enough information to prove that JKLM is an isosceles trapezoid. Explain. M L J GIVEN c }JK i}LM , JKL > KJM

TRAPEZOIDS In Exercises 2931, determine whether there is enough information to prove that JKLM is an isosceles trapezoid. Explain. M L J GIVEN c }JK i}LM , JML > KLM, m KLM 908

TRAPEZOIDS In Exercises 2931, determine whether there is enough information to prove that JKLM is an isosceles trapezoid. Explain. M L J GIVEN c }JL >}KM , }JK i}LM , JK > LM

CHALLENGE Draw a rectangle and bisect its angles. What type of quadrilateral is formed by the intersecting bisectors? Justify your answer.

REAL-WORLD OBJECTS What type of special quadrilateral is outlined?

REAL-WORLD OBJECTS What type of special quadrilateral is outlined? x x x 1 2 x 1 2

REAL-WORLD OBJECTS What type of special quadrilateral is outlined?

PYRAMID Use the photo of the Pyramid of Kukulcan in Mexico. a. EF}i}HG , and }EH and }FG are not parallel. What shape is this part of the pyramid? b. AB}i }DC , }AD i }BC , and A, B, C, and D are all congruent to each other. What shape is this part of the pyramid? E H G F A

SHORT RESPONSE Explain why a parallelogram with one right angle must be a rectangle.

EXTENDED RESPONSE Segments AC and BD bisect each other. a. Suppose that }AC and }BD are congruent, but not perpendicular. Draw quadrilateral ABCD and classify it. Justify your answer. b. Suppose that }AC and }BD are perpendicular, but not congruent. Draw quadrilateral ABCD and classify it. Justify your answer.

MULTI-STEP PROBLEM Polygon QRSTUV shown at the right is a regular hexagon, and}QU and }RT are diagonals. Follow the steps below to classify quadrilateral QRTU. Explain your reasoning in each step. a. Show that nQVU and nRST are congruent isosceles triangles. b. Show that }QR > }UT and that}QU > }RT . c. Show that UQR > QRT > RTU > TUQ. Find the measure of each of these angles. d. Classify quadrilateral QRTU. U T P

REASONING In quadrilateral WXYZ,}WY and }XZ intersect each other at point V.}WV > }XV and }YV > }ZV , but}WY and }XZ do not bisect each other. Draw}WY , }XY , and WXYZ. What special type of quadrilateral is WXYZ? Write a plan for a proof of your answer.

CHALLENGE What special type of quadrilateral is EFGH? Write a paragraph proof to show that your answer is correct. GIVEN c PQRS is a square. E, F, G, and H are midpoints of the sides of the square. PROVE c EFGH is a ? R L G S E H P

CHALLENGE What special type of quadrilateral is EFGH? Write a paragraph proof to show that your answer is correct. GIVEN c In the three-dimensional figure, JK}>}LM ; E, F, G, and H are the midpoints of }JL , }KL ,}KM , and }JM . PROVE c EFGH is a ? M G H F E K

In Exercises 43 and 44, use the diagram. (p. 264) B y x 1 3 7 Find the values of x and y. Explain your reasoning.

In Exercises 43 and 44, use the diagram. (p. 264) B y x 1 3 7 Find m ADC, m DAC, and m DCA. Explain your reasoning.

The vertices of quadrilateral ABCD are A(22, 1), B(2, 5), C(3, 2), and D(1, 21). Draw ABCD in a coordinate plane. Then draw its image after the indicated translation. (p. 272) (x, y) (x 1 1, y 2 3)

The vertices of quadrilateral ABCD are A(22, 1), B(2, 5), C(3, 2), and D(1, 21). Draw ABCD in a coordinate plane. Then draw its image after the indicated translation. (p. 272) (x, y) (x 2 2, y 2 2)

Use the diagram of ~WXYZ to find the indicated length. (p. 515) V 4 12 5 Z W YZ

Use the diagram of ~WXYZ to find the indicated length. (p. 515) V 4 12 5 Z W WZ

Use the diagram of ~WXYZ to find the indicated length. (p. 515) V 4 12 5 Z W XV

Use the diagram of ~WXYZ to find the indicated length. (p. 515) V 4 12 5 Z W XZ

In Exercises 1 and 2, copy and complete the statement. The ? of a trapezoid is parallel to the bases.

In Exercises 1 and 2, copy and complete the statement. A(n) ? of a polygon is a segment whose endpoints are nonconsecutive vertices.

WRITING Describe the different ways you can show that a trapezoid is an isosceles trapezoid.

In Exercises 46, match the figure with the most specific name. A. Square

In Exercises 46, match the figure with the most specific name. B. Parallelogram

In Exercises 46, match the figure with the most specific name. C. Rhombus

The sum of the measures of the interior angles of a convex regular polygon is 39608. Classify the polygon by the number of sides. What is the measure of each interior angle?

In Exercises 810, find the value of x. 1208 x8 978 1308 150

In Exercises 810, find the value of x. x8 2x8 1258 1108 1478 1128

In Exercises 810, find the value of x. 5x8 5x8 8x

In a regular nonagon, the exterior angles are all congruent. What is the measure of one of the exterior angles? Explain.

Find the value of each variable in the parallelogram. 10 8 n 2 3 m

Find the value of each variable in the parallelogram. 11 14 c 1 5 d 1 4

Find the value of each variable in the parallelogram. 18 a 2 10 (b 1 16)8 1038

In ~PQRS, PQ 5 5 centimeters, QR 5 10 centimeters, and m PQR 5 368. Sketch PQRS. Find and label all of its side lengths and interior angle measures.

The perimeter of ~EFGH is 16 inches. If EF is 5 inches, find the lengths of all the other sides of EFGH. Explain your reasoning.

In ~JKLM, the ratio of the measure of J to the measure of M is 5 : 4. Find m J and m M. Explain your reasoning.

For what value of x is the quadrilateral a parallelogram? 2x 1 4 x 1 9

For what value of x is the quadrilateral a parallelogram? 3x 1 2 5x 2 4

Classify the special quadrilateral. Then find the values of x and y. y8 x8

Classify the special quadrilateral. Then find the values of x and y. 3x 1 4 4y 6y 2 10

The diagonals of a rhombus are 10 centimeters and 24 centimeters. Find the length of a side. Explain.

In Exercises 23 and 24, use the diagram of a recycling container. One end of the container is an isosceles trapezoid with}FG i }JH and m F 5 798. J H F Find m G, m H, and m J.

In Exercises 23 and 24, use the diagram of a recycling container. One end of the container is an isosceles trapezoid with}FG i }JH and m F 5 798. J H F Copy trapezoid FGHJ and sketch its midsegment. If the midsegment is 16.5 inches long and }FG is 19 inches long, find JH.

Give the most specific name for the quadrilateral. Explain your reasoning. D C A

Give the most specific name for the quadrilateral. Explain your reasoning. E F 1488 328

Give the most specific name for the quadrilateral. Explain your reasoning. J K 10 10 7 7

In quadrilateral RSTU, R, T, and U are right angles, and RS 5 ST. What is the most specific name for quadrilateral RSTU? Explain.