 4.4.2.1: VOCABULARY Copy the congruent triangles shown. Then label the verti...
 4.4.8.1: VOCABULARY Describe the translation (x, y) (x 2 1, y 1 4) in words.
 4.4.3.1: VOCABULARY Tell whether the angles or sides are corresponding angle...
 4.1: Copy and complete: A triangle with three congruent angles is called...
 4.4.4.1: VOCABULARY Copy and complete: The angle between two sides of a tria...
 4.4.5.1: VOCABULARY Name one advantage of using a flow proof rather than a t...
 4.4.6.1: VOCABULARY Copy and complete: Corresponding parts of congruent tria...
 4.4.1.1: VOCABULARY Match the triangle description with the most specific na...
 4.4.7.1: VOCABULARY Define the vertex angle of an isosceles triangle.
 4.4.2.2: WRITING Based on this lesson, what information do you need to prove...
 4.4.8.2: WRITING Explain why the term congruence transformation is used in d...
 4.4.3.2: VOCABULARY Tell whether the angles or sides are corresponding angle...
 4.2: WRITING Compare vertex angles and base angles.
 4.4.4.2: WRITING Explain the difference between proving triangles congruent ...
 4.4.5.2: WRITING You know that a pair of triangles has two pairs of congruen...
 4.4.6.2: WRITING Explain why you might choose to use congruent triangles to ...
 4.4.1.2: VOCABULARY Match the triangle description with the most specific na...
 4.4.7.2: WRITING What is the relationship between the base angles of an isos...
 4.4.2.3: USING CONGRUENCE Identify all pairs of congruent corresponding part...
 4.4.8.3: IDENTIFYING TRANSFORMATIONS Name the type of transformation shown. y24
 4.4.3.3: VOCABULARY Tell whether the angles or sides are corresponding angle...
 4.3: WRITING Describe the difference between isosceles and scalene trian...
 4.4.4.3: NAMING INCLUDED ANGLES Use the diagram to name the included angle b...
 4.4.5.3: IDENTIFY CONGRUENT TRIANGLES Is it possible to prove that the trian...
 4.4.6.3: CONGRUENT TRIANGLES Tell which triangles you can show are congruent...
 4.4.1.3: VOCABULARY Match the triangle description with the most specific na...
 4.4.7.3: USING DIAGRAMS In Exercises 36, use the diagram. Copy and complete ...
 4.4.7.4: USING DIAGRAMS In Exercises 36, use the diagram. Copy and complete ...
 4.4.2.4: USING CONGRUENCE Identify all pairs of congruent corresponding part...
 4.4.8.4: IDENTIFYING TRANSFORMATIONS Name the type of transformation shown. xy
 4.4.3.4: VOCABULARY Tell whether the angles or sides are corresponding angle...
 4.4: Sketch an acute scalene triangle. Label its interior angles 1, 2, a...
 4.4.4.4: NAMING INCLUDED ANGLES Use the diagram to name the included angle b...
 4.4.5.4: IDENTIFY CONGRUENT TRIANGLES Is it possible to prove that the trian...
 4.4.6.4: CONGRUENT TRIANGLES Tell which triangles you can show are congruent...
 4.4.1.4: VOCABULARY Match the triangle description with the most specific na...
 4.4.7.5: USING DIAGRAMS In Exercises 36, use the diagram. Copy and complete ...
 4.4.2.5: READING A DIAGRAM In the diagram, nXYZ > nMNL. Copy and complete th...
 4.4.8.5: IDENTIFYING TRANSFORMATIONS Name the type of transformation shown. y21
 4.4.3.5: DETERMINING CONGRUENCE Decide whether the congruence statement is t...
 4.5: If nPQR > nLMN, which angles are corresponding angles? Which sides ...
 4.4.4.5: NAMING INCLUDED ANGLES Use the diagram to name the included angle b...
 4.4.5.5: IDENTIFY CONGRUENT TRIANGLES Is it possible to prove that the trian...
 4.4.6.5: CONGRUENT TRIANGLES Tell which triangles you can show are congruent...
 4.4.1.5: VOCABULARY Match the triangle description with the most specific na...
 4.4.7.6: USING DIAGRAMS In Exercises 36, use the diagram. Copy and complete ...
 4.4.2.6: READING A DIAGRAM In the diagram, nXYZ > nMNL. Copy and complete th...
 4.4.8.6: WINDOWS Decide whether the moving part of the window is a translati...
 4.4.3.6: DETERMINING CONGRUENCE Decide whether the congruence statement is t...
 4.6: Find the measure of the exterior angle shown. 2x 2 25)8x8 208
 4.4.4.6: NAMING INCLUDED ANGLES Use the diagram to name the included angle b...
 4.4.5.6: ERROR ANALYSIS Describe the error in concluding that nABC > nXYZ. B...
 4.4.6.6: CONGRUENT TRIANGLES Tell which triangles you can show are congruent...
 4.4.1.6: VOCABULARY Match the triangle description with the most specific na...
 4.4.7.7: REASONING Find the unknown measure. AB C? 12
 4.4.2.7: READING A DIAGRAM In the diagram, nXYZ > nMNL. Copy and complete th...
 4.4.8.7: WINDOWS Decide whether the moving part of the window is a translati...
 4.4.3.7: DETERMINING CONGRUENCE Decide whether the congruence statement is t...
 4.7: Find the measure of the exterior angle shown. 8x82
 4.4.4.7: NAMING INCLUDED ANGLES Use the diagram to name the included angle b...
 4.4.5.7: MULTIPLE CHOICE Which postulate or theorem can you use to prove tha...
 4.4.6.7: CONGRUENT TRIANGLES Tell which triangles you can show are congruent...
 4.4.1.7: WRITING Can a right triangle also be obtuse? Explain why or why not.
 4.4.7.8: REASONING Find the unknown measure. ML 16 N?608 608
 4.4.2.8: READING A DIAGRAM In the diagram, nXYZ > nMNL. Copy and complete th...
 4.4.8.8: WINDOWS Decide whether the moving part of the window is a translati...
 4.4.3.8: ERROR ANALYSIS Describe and correct the error in writing a congruen...
 4.8: Find the measure of the exterior angle shown. 458 5x8(9
 4.4.4.8: NAMING INCLUDED ANGLES Use the diagram to name the included angle b...
 4.4.5.8: DEVELOPING PROOF State the third congruence that is needed to prove...
 4.4.6.8: CONGRUENT TRIANGLES Tell which triangles you can show are congruent...
 4.4.1.8: CLASSIFYING TRIANGLES Copy the triangle and measure its angles. Cla...
 4.4.7.9: REASONING Find the unknown measure. SR T?
 4.4.2.9: READING A DIAGRAM In the diagram, nXYZ > nMNL. Copy and complete th...
 4.4.8.9: DRAWING A TRANSLATION Copy figure ABCD and draw its image after the...
 4.4.3.9: ALGEBRA Use the given coordinates to determine if nABC > nDEF. A(22...
 4.9: In the diagram, nABC > nVTU. Find the indicated measure. A C TU V15...
 4.4.4.9: REASONING Decide whether enough information is given to prove that ...
 4.4.5.9: DEVELOPING PROOF State the third congruence that is needed to prove...
 4.4.6.9: ERROR ANALYSIS Describe the error in the statement. ABC > nCDA by S...
 4.4.1.9: CLASSIFYING TRIANGLES Copy the triangle and measure its angles. Cla...
 4.4.7.10: DRAWING DIAGRAMS A base angle in an isosceles triangle measures 378...
 4.4.2.10: READING A DIAGRAM In the diagram, nXYZ > nMNL. Copy and complete th...
 4.4.8.10: DRAWING A TRANSLATION Copy figure ABCD and draw its image after the...
 4.4.3.10: ALGEBRA Use the given coordinates to determine if nABC > nDEF. A(22...
 4.10: In the diagram, nABC > nVTU. Find the indicated measure. A C TU V15...
 4.4.4.10: REASONING Decide whether enough information is given to prove that ...
 4.4.5.10: DEVELOPING PROOF State the third congruence that is needed to prove...
 4.4.6.10: PLANNING FOR PROOF Use the diagram to write a plan for proof. PROVE...
 4.4.1.10: CLASSIFYING TRIANGLES Copy the triangle and measure its angles. Cla...
 4.4.7.11: ALGEBRA Find the value of x. 5553x 8
 4.4.2.11: NAMING CONGRUENT FIGURES Write a congruence statement for any figur...
 4.4.8.11: DRAWING A TRANSLATION Copy figure ABCD and draw its image after the...
 4.4.3.11: ALGEBRA Use the given coordinates to determine if nABC > nDEF. A(0,...
 4.11: In the diagram, nABC > nVTU. Find the indicated measure. A C TU V15...
 4.4.4.11: REASONING Decide whether enough information is given to prove that ...
 4.4.5.11: OVERLAPPING TRIANGLES Explain how you can prove that the indicated ...
 4.4.6.11: PLANNING FOR PROOF Use the diagram to write a plan for proof. PROVE...
 4.4.1.11: COORDINATE PLANE A triangle has the given vertices. Graph the trian...
 4.4.7.12: ALGEBRA Find the value of x. K165x 1 535
 4.4.2.12: NAMING CONGRUENT FIGURES Write a congruence statement for any figur...
 4.4.8.12: DRAWING A TRANSLATION Copy figure ABCD and draw its image after the...
 4.4.3.12: ALGEBRA Use the given coordinates to determine if nABC > nDEF. A(25...
 4.12: In the diagram, nABC > nVTU. Find the indicated measure. A C TU V15...
 4.4.4.12: REASONING Decide whether enough information is given to prove that ...
 4.4.5.12: OVERLAPPING TRIANGLES Explain how you can prove that the indicated ...
 4.4.6.12: PENTAGONS Explain why segments connecting any pair of corresponding...
 4.4.1.12: COORDINATE PLANE A triangle has the given vertices. Graph the trian...
 4.4.7.13: ALGEBRA Find the value of x. B7289x8
 4.4.2.13: NAMING CONGRUENT FIGURES Write a congruence statement for any figur...
 4.4.8.13: COORDINATE NOTATION Use coordinate notation to describe the transla...
 4.4.3.13: USING DIAGRAMS Decide whether the figure is stable. Explain.
 4.13: Find the value of x. 1208 208(2x 1 4)8
 4.4.4.13: REASONING Decide whether enough information is given to prove that ...
 4.4.5.13: OVERLAPPING TRIANGLES Explain how you can prove that the indicated ...
 4.4.6.13: ALGEBRA Given that nABC > nDEF, mA 5 708, mB 5 608, mC 5 508, mD 5 ...
 4.4.1.13: COORDINATE PLANE A triangle has the given vertices. Graph the trian...
 4.4.7.14: ERROR ANALYSIS Describe and correct the error made in finding BC in...
 4.4.2.14: NAMING CONGRUENT FIGURES Write a congruence statement for any figur...
 4.4.8.14: COORDINATE NOTATION Use coordinate notation to describe the transla...
 4.4.3.14: USING DIAGRAMS Decide whether the figure is stable. Explain.
 4.14: Find the value of x. 3585x
 4.4.4.14: REASONING Decide whether enough information is given to prove that ...
 4.4.5.14: DETERMINING CONGRUENCE Tell whether you can use the given informati...
 4.4.6.14: MULTIPLE CHOICE Which set of given information does not allow you t...
 4.4.1.14: FINDING ANGLE MEASURES Find the value of x. Then classify the trian...
 4.4.7.15: ALGEBRA Find the values of x and y. x8y8102
 4.4.2.15: THIRD ANGLES THEOREM Find the value of x. LNM Y708x8
 4.4.8.15: COORDINATE NOTATION Use coordinate notation to describe the transla...
 4.4.3.15: USING DIAGRAMS Decide whether the figure is stable. Explain.
 4.15: Decide whether the congruence statement is true. Explain your reaso...
 4.4.4.15: MULTIPLE CHOICE Which of the following sets of information does not...
 4.4.5.15: DETERMINING CONGRUENCE Tell whether you can use the given informati...
 4.4.6.15: PLANNING FOR PROOF Use the information given in the diagram to writ...
 4.4.1.15: FINDING ANGLE MEASURES Find the value of x. Then classify the trian...
 4.4.7.16: ALGEBRA Find the values of x and y. (x 1 7)8y
 4.4.2.16: THIRD ANGLES THEOREM Find the value of x. A CRB S P4585x8 808
 4.4.8.16: COORDINATE NOTATION Use coordinate notation to describe the transla...
 4.4.3.16: MULTIPLE CHOICE Let nFGH be an equilateral triangle with point J as...
 4.16: Decide whether the congruence statement is true. Explain your reaso...
 4.4.4.16: APPLYING SAS In Exercises 1618, use the given information to name t...
 4.4.5.16: DETERMINING CONGRUENCE Tell whether you can use the given informati...
 4.4.6.16: PLANNING FOR PROOF Use the information given in the diagram to writ...
 4.4.1.16: FINDING ANGLE MEASURES Find the value of x. Then classify the trian...
 4.4.7.17: ALGEBRA Find the values of x and y. 9y8x
 4.4.2.17: ERROR ANALYSIS A student says that nMNP > nRSP because the correspo...
 4.4.8.17: DRAWING Use a reflection in the xaxis to draw the other half of th...
 4.4.3.17: MULTIPLE CHOICE Let ABCD be a rectangle separated into two triangle...
 4.17: Decide whether the congruence statement is true. Explain your reaso...
 4.4.4.17: APPLYING SAS In Exercises 1618, use the given information to name t...
 4.4.5.17: DETERMINING CONGRUENCE Tell whether you can use the given informati...
 4.4.6.17: PLANNING FOR PROOF Use the information given in the diagram to writ...
 4.4.1.17: ALGEBRA Find the measure of the exterior angle shown. (2x 2 2)8x8 45
 4.4.7.18: SHORT RESPONSE Are isosceles triangles always acute triangles? Expl...
 4.4.2.18: OPENENDED MATH Graph the triangle with vertices L(3, 1), M(8, 1), ...
 4.4.8.18: DRAWING Use a reflection in the xaxis to draw the other half of th...
 4.4.3.18: APPLYING SEGMENT ADDITION Determine whether nABC > nDEF. If they ar...
 4.18: Decide whether the congruence statement is true. Explain your reaso...
 4.4.4.18: APPLYING SAS In Exercises 1618, use the given information to name t...
 4.4.5.18: IDENTIFY CONGRUENT TRIANGLES Is it possible to prove that the trian...
 4.4.6.18: PLANNING FOR PROOF Use the information given in the diagram to writ...
 4.4.1.18: ALGEBRA Find the measure of the exterior angle shown. (2x 1 18)8 (3...
 4.4.7.19: MULTIPLE CHOICE What is the value of x in the diagram? A 5 B 6 C 7 ...
 4.4.2.19: ALGEBRA Find the values of x and y (17x 2 y)8(12x 1 4y)8
 4.4.8.19: DRAWING Use a reflection in the xaxis to draw the other half of th...
 4.4.3.19: APPLYING SEGMENT ADDITION Determine whether nABC > nDEF. If they ar...
 4.19: State the third congruence that is needed to prove that nDEF > nGHJ...
 4.4.4.19: OVERLAPPING TRIANGLES Redraw nACF and nEGB so they are side by side...
 4.4.5.19: IDENTIFY CONGRUENT TRIANGLES Is it possible to prove that the trian...
 4.4.6.19: PLANNING FOR PROOF Use the information given in the diagram to writ...
 4.4.1.19: ALGEBRA Find the measure of the exterior angle shown. (3x 1 2)8x
 4.4.7.20: ALGEBRA Find the values of x and y, if possible. Explain your reaso...
 4.4.2.20: ALGEBRA Find the values of x and y 4x 1 y)8 (6x 2 y)
 4.4.8.20: ROTATIONS Use the coordinates to graph}AB and}CD . Tell whether}CD ...
 4.4.3.20: 3D FIGURES In the diagram, }PK > }PL and }JK > }JL . Show that nJP...
 4.20: State the third congruence that is needed to prove that nDEF > nGHJ...
 4.4.4.20: REASONING Decide whether enough information is given to prove that ...
 4.4.5.20: IDENTIFY CONGRUENT TRIANGLES Is it possible to prove that the trian...
 4.4.6.20: PLANNING FOR PROOF Use the information given in the diagram to writ...
 4.4.1.20: SHORT RESPONSE Explain how to use the Corollary to the Triangle Sum...
 4.4.7.21: ALGEBRA Find the values of x and y, if possible. Explain your reaso...
 4.4.2.21: MULTIPLE CHOICE Suppose nABC > nEFD, nEFD > nGIH, mA 5 908, and mF ...
 4.4.8.21: ROTATIONS Use the coordinates to graph}AB and}CD . Tell whether}CD ...
 4.4.3.21: CHALLENGE Find all values of x that make the triangles congruent. E...
 4.21: Write a plan for proving that 1 > 2. A BC D E1 2
 4.4.4.21: REASONING Decide whether enough information is given to prove that ...
 4.4.5.21: EXTENDED RESPONSE Use the graph at the right. yD(4, 2)A(0, 1)C(6, 6...
 4.4.6.21: USING COORDINATES Use the vertices of nABC and nDEF to show that A ...
 4.4.1.21: ANGLE RELATIONSHIPS Find the measure of the numbered angle. 408 208 1
 4.4.7.22: ALGEBRA Find the values of x and y, if possible. Explain your reaso...
 4.4.2.22: CHALLENGE A hexagon is contained in a cube, as shown. Each vertex o...
 4.4.8.22: ROTATIONS Use the coordinates to graph}AB and}CD . Tell whether}CD ...
 4.4.3.22: TILE FLOORS You notice two triangles in the tile floor of a hotel l...
 4.22: Write a plan for proving that 1 > 2. FK G
 4.4.4.22: REASONING Decide whether enough information is given to prove that ...
 4.4.5.22: CHALLENGE Use a coordinate plane. a. Graph the lines y 5 2x 1 5, y ...
 4.4.6.22: USING COORDINATES Use the vertices of nABC and nDEF to show that A ...
 4.4.1.22: ANGLE RELATIONSHIPS Find the measure of the numbered angle. 408 209 2
 4.4.7.23: ALGEBRA Find the perimeter of the triangle. (x 1 3) ft (2x 1 1) ft6 ft
 4.4.2.23: RUG DESIGNS The rug design is made of congruent triangles. One tria...
 4.4.8.23: ROTATIONS Use the coordinates to graph}AB and}CD . Tell whether}CD ...
 4.4.3.23: GATES Which gate is stable? Explain your reasoning.
 4.23: Write a plan for proving that 1 > 2. PRVS T1 2
 4.4.4.23: WRITING Suppose both pairs of corresponding legs of two right trian...
 4.4.5.23: CONGRUENCE IN BICYCLES Explain why the triangles are congruent.
 4.4.6.23: PROOF Use the information given in the diagram to write a proof. PR...
 4.4.1.23: ANGLE RELATIONSHIPS Find the measure of the numbered angle. 408 210 3
 4.4.7.24: ALGEBRA Find the perimeter of the triangle. (x 1 4) in.(4x 1 1) in
 4.4.2.24: OPENENDED MATH Create a design for a rug made with congruent trian...
 4.4.8.24: ERROR ANALYSIS A student says that the red triangle is a 1208 clock...
 4.4.3.24: PROOF Write a proof. GIVEN c }GH > }JK , }HJ > }KG PROVE c nGHJ > n...
 4.24: Find the value of x. LMNx8658
 4.4.4.24: ERROR ANALYSIS Describe and correct the error in finding the value ...
 4.4.5.24: CONGRUENCE IN BICYCLES Explain why the triangles are congruent.
 4.4.6.24: PROOF Use the information given in the diagram to write a proof. PR...
 4.4.1.24: ANGLE RELATIONSHIPS Find the measure of the numbered angle. 408 211 4
 4.4.1.25: ANGLE RELATIONSHIPS Find the measure of the numbered angle. 408 212 5
 4.4.7.25: ALGEBRA Find the perimeter of the triangle. (21 2 x) in.(2x 2 3) in...
 4.4.2.25: CAR STEREO A car stereo fits into a space in your dashboard. You wa...
 4.4.8.25: WRITING Can a point or a line segment be its own image under a tran...
 4.4.3.25: PROOF Write a proof. GIVEN c }WX > }VZ ,}WY > }VY , }YZ > }YX PROVE...
 4.25: Find the value of x. WYX9 ft9 ft9 ft X x 1 30C8
 4.4.4.25: USING DIAGRAMS In Exercises 2527, state the third congruence that m...
 4.4.5.25: FLOW PROOF Copy and complete the flow proof. GIVEN c }AD i }CE , }B...
 4.4.6.25: PROOF Use the information given in the diagram to write a proof. PR...
 4.4.1.26: ANGLE RELATIONSHIPS Find the measure of the numbered angle. 408 213 6
 4.4.7.26: REASONING In Exercises 2629, use the diagram. State whether the giv...
 4.4.2.26: PROOF Copy and complete the proof. GIVEN c }AB > }ED , }BC > }DC , ...
 4.4.8.26: APPLYING TRANSLATIONS Complete the statement using the description ...
 4.4.3.26: PROOF Write a proof. GIVEN c }AE > }CE , }AB > }CD , E is the midpo...
 4.26: Find the value of x. TUV1013 2 x7x 1 5
 4.4.4.26: USING DIAGRAMS In Exercises 2527, state the third congruence that m...
 4.4.5.26: SHORT RESPONSE You are making a map for an orienteering race. Parti...
 4.4.6.26: PROOF Use the information given in the diagram to write a proof. PR...
 4.4.1.27: ALGEBRA In nPQR, P > R and the measure of Q is twice the measure of...
 4.4.7.27: REASONING In Exercises 2629, use the diagram. State whether the giv...
 4.4.2.27: SHORT RESPONSE Suppose nABC > nDCB, and the triangles share vertice...
 4.4.8.27: APPLYING TRANSLATIONS Complete the statement using the description ...
 4.4.3.27: PROOF Write a proof. GIVEN c }FM > }FN ,}DM >}HN EF}> }GF , }DE >}H...
 4.27: Triangle QRS has vertices Q(2, 21), R(5, 22), and S(2, 23). Sketch ...
 4.4.4.27: USING DIAGRAMS In Exercises 2527, state the third congruence that m...
 4.4.5.27: AIRPLANE In the airplane at the right, C and F are right angles, BC...
 4.4.6.27: CHALLENGE Which of the triangles below are congruent? CA BPN PFD EJ...
 4.4.1.28: ALGEBRA In nEFG, m F 5 3(m G), and m E 5 m F 2 308. Find the measur...
 4.4.7.28: REASONING In Exercises 2629, use the diagram. State whether the giv...
 4.4.2.28: PROVING THEOREM 4.3 Use the plan to prove the Third Angles Theorem....
 4.4.8.28: APPLYING TRANSLATIONS Complete the statement using the description ...
 4.4.3.28: EXTENDED RESPONSE When rescuers enter a partially collapsed buildin...
 4.28: Triangle QRS has vertices Q(2, 21), R(5, 22), and S(2, 23). Sketch ...
 4.4.4.28: USING ISOSCELES TRIANGLES Suppose nKLN and nMLN are isosceles trian...
 4.4.5.28: RIGHT TRIANGLES In Lesson 4.4, you learned the HypotenuseLeg Theor...
 4.4.6.28: CANYON Explain how you can find the distance across the canyon. DA BCE
 4.4.1.29: ERROR ANALYSIS In Exercises 29 and 30, describe and correct the err...
 4.4.7.29: REASONING In Exercises 2629, use the diagram. State whether the giv...
 4.4.2.29: REASONING Given that nAFC > nDFE, must F be the midpoint of }AD and...
 4.4.8.29: ALGEBRA A point on an image and the translation are given. Find the...
 4.4.3.29: BASEBALL FIELD To create a baseball field, start by placing home pl...
 4.29: Triangle QRS has vertices Q(2, 21), R(5, 22), and S(2, 23). Sketch ...
 4.4.4.29: REASONING Suppose M is the midpoint of }PQ in nPQR. If }RM }PQ , ex...
 4.4.5.29: RIGHT TRIANGLES In Lesson 4.4, you learned the HypotenuseLeg Theor...
 4.4.6.29: PROOF Use the given information and the diagram to write a twocolu...
 4.4.1.30: ERROR ANALYSIS In Exercises 29 and 30, describe and correct the err...
 4.4.7.30: SHORT RESPONSE In nDEF, m D 5 (4x 1 2)8, m E 5 (6x 2 30)8, and m F ...
 4.4.2.30: SHORT RESPONSE You have a set of tiles that come in two different s...
 4.4.8.30: ALGEBRA A point on an image and the translation are given. Find the...
 4.4.3.30: CHALLENGE Draw and label the figure described below. Then, identify...
 4.4.4.30: CHALLENGE Suppose }AB > }AC , }AD > }AF , }AD }AB , and }AF }AC . E...
 4.4.5.30: RIGHT TRIANGLES In Lesson 4.4, you learned the HypotenuseLeg Theor...
 4.4.6.30: SNOWBOARDING In the diagram of the half pipe below, C is the midpoi...
 4.4.1.31: MULTIPLE CHOICE Which of the following is not possible? A An acute ...
 4.4.7.31: SHORT RESPONSE In nABC, D is the midpoint of }AC , and }BD is perpe...
 4.4.2.31: MULTISTEP PROBLEM In the diagram, quadrilateral ABEF > quadrilater...
 4.4.8.31: ALGEBRA A point on an image and the translation are given. Find the...
 4.4.3.31: Find the slope of the line that passes through the points. (p. 171)...
 4.4.4.31: CONGRUENT TRIANGLES In Exercises 31 and 32, identify the theorem or...
 4.4.5.31: PROOF Write a twocolumn proof. GIVEN c }AK > }CJ , BJK > BKJ, A > ...
 4.4.6.31: MULTIPLE CHOICE Using the information in the diagram, you can prove...
 4.4.1.32: ALGEBRA In Exercises 3237, find the values of x and y. 7584
 4.4.7.32: ALGEBRA Find the value(s) of the variable(s). Explain your reasonin...
 4.4.2.32: CHALLENGE Use the diagram to write a proof. GIVEN c }WX ] VZ at Y, ...
 4.4.8.32: CONGRUENCE Show that the transformation in Exercise 3 is a congruen...
 4.4.3.32: Find the slope of the line that passes through the points. (p. 171)...
 4.4.4.32: CONGRUENT TRIANGLES In Exercises 31 and 32, identify the theorem or...
 4.4.5.32: PROOF Write a flow proof. GIVEN c }VW >}UW , X > Z PROVE c nXWV > n...
 4.4.6.32: PROVING A CONSTRUCTION The diagrams below show the construction on ...
 4.4.1.33: ALGEBRA In Exercises 3237, find the values of x and y. 1188y8
 4.4.7.33: ALGEBRA Find the value(s) of the variable(s). Explain your reasonin...
 4.4.2.33: Use the Distance Formula to find the length of the segment. Round y...
 4.4.8.33: DESCRIBING AN IMAGE State the segment or triangle that represents t...
 4.4.3.33: Find the slope of the line that passes through the points. (p. 171)...
 4.4.4.33: SAILBOATS Suppose you have two sailboats. What information do you n...
 4.4.5.33: PROOF Write a proof. GIVEN c NKM > LMK, L > N PROVE c nNMK > nLKM K...
 4.4.6.33: ARCHITECTURE Can you use the given information to determine that }A...
 4.4.1.34: ALGEBRA In Exercises 3237, find the values of x and y. x82082
 4.4.7.34: ALGEBRA Find the value(s) of the variable(s). Explain your reasonin...
 4.4.2.34: Use the Distance Formula to find the length of the segment. Round y...
 4.4.8.34: DESCRIBING AN IMAGE State the segment or triangle that represents t...
 4.4.3.34: Use the x and yintercepts to write an equation of the line. (p. 1...
 4.4.4.34: DEVELOPING PROOF Copy and complete the proof. GIVEN c Point M is th...
 4.4.5.34: PROOF Write a proof. GIVEN c X is the midpoint of}VY and}WZ PROVE c...
 4.4.6.34: ARCHITECTURE Can you use the given information to determine that }A...
 4.4.1.35: ALGEBRA In Exercises 3237, find the values of x and y. x8 648 y
 4.4.7.35: REASONING The measure of an exterior angle of an isosceles triangle...
 4.4.2.35: Use the Distance Formula to find the length of the segment. Round y...
 4.4.8.35: DESCRIBING AN IMAGE State the segment or triangle that represents t...
 4.4.3.35: Use the x and yintercepts to write an equation of the line. (p. 1...
 4.4.4.35: PROOF In Exercises 35 and 36, write a proof. GIVEN c }PQ bisects SP...
 4.4.5.35: CHALLENGE Write a proof. GIVEN c nABF > nDFB, F is the midpoint of ...
 4.4.6.35: ARCHITECTURE Can you use the given information to determine that }A...
 4.4.1.36: ALGEBRA In Exercises 3237, find the values of x and y. x8478158y8
 4.4.7.36: PROOF Let nABC be isosceles with vertex angle A. Suppose A, B, and ...
 4.4.2.36: Line l bisects the segment. Write a congruence statement. (p. 15) l...
 4.4.8.36: DESCRIBING AN IMAGE State the segment or triangle that represents t...
 4.4.3.36: Use the x and yintercepts to write an equation of the line. (p. 1...
 4.4.4.36: PROOF In Exercises 35 and 36, write a proof. GIVEN c }VX > }XY ,}XW...
 4.4.5.36: Find the value of x that makes m i n. (p. 161) 518x8nm
 4.4.6.36: EXTENDED RESPONSE You can use the method described below to find th...
 4.4.1.37: ALGEBRA In Exercises 3237, find the values of x and y. 358188y8
 4.4.7.37: CHALLENGE The measure of an exterior angle of an isosceles triangle...
 4.4.2.37: Line l bisects the segment. Write a congruence statement. (p. 15) lL N
 4.4.8.37: CHALLENGE Solve for the variables in the transformation of }AB to }...
 4.4.3.37: Write an equation of a line that passes through (23, 21) and is par...
 4.4.4.37: PROOF In Exercises 37 and 38, write a proof. GIVEN c }JM >}LM PROVE...
 4.4.5.37: Find the value of x that makes m i n. (p. 161) 4282x8nm
 4.4.6.37: PROOF Use the given information and the diagram to prove that 1 > 2...
 4.4.1.38: VISUALIZATION Is there an angle measure that is so small that any t...
 4.4.7.38: SPORTS The dimensions of a sports pennant are given in the diagram....
 4.4.2.38: Line l bisects the segment. Write a congruence statement. (p. 15) R...
 4.4.8.38: KITES The design for a kite shows the layout and dimensions for onl...
 4.4.4.38: PROOF In Exercises 37 and 38, write a proof. GIVEN c D is the midpo...
 4.4.5.38: Find the value of x that makes m i n. (p. 161) 1018(x 1 16)8 nm
 4.4.6.38: PROOF Use the given information and the diagram to prove that 1 > 2...
 4.4.1.39: CHALLENGE Suppose you have the equations y 5 ax 1 b, y 5 cx 1 d, an...
 4.4.7.39: ADVERTISING A logo in an advertisement is an equilateral triangle w...
 4.4.2.39: Write the converse of the statement. (p. 79) . If three points are ...
 4.4.8.39: STENCILING You are stenciling a room in your home. You want to use ...
 4.4.4.39: MULTIPLE CHOICE Which triangle congruence can you prove, then use t...
 4.4.5.39: Write an equation of the line that passes through point P and is pa...
 4.4.6.39: PROOF Write a proof. GIVEN c }BA > }BC, D and E are midpoints, A > ...
 4.4.1.40: THEATER Three people are standing on a stage. The distances between...
 4.4.7.40: ARCHITECTURE The Transamerica Pyramid building shown in the photogr...
 4.4.2.40: Write the converse of the statement. (p. 79) If the sky is cloudy, ...
 4.4.8.40: OPENENDED MATH Some words reflect onto themselves through a vertic...
 4.4.4.40: PROOF Write a twocolumn proof. GIVEN c }CR > }CS , }QC }CR , }QC }...
 4.4.5.40: Write an equation of the line that passes through point P and is pa...
 4.4.6.40: CHALLENGE In the diagram of pentagon ABCDE, }AB i }EC , }AC i }ED ,...
 4.4.1.41: KALEIDOSCOPES You are making a kaleidoscope. The directions state t...
 4.4.7.41: MULTISTEP PROBLEM To make a zigzag pattern, a graphic designer sk...
 4.4.8.41: SHORT RESPONSE In chess, six different kinds of pieces are moved ac...
 4.4.4.41: CHALLENGE Describe how to show that nPMO > nPMN using the SSS Congr...
 4.4.5.41: Decide which method, SSS, SAS, or HL, can be used to prove that the...
 4.4.6.41: How many lines can be drawn that fit each description? Copy the dia...
 4.4.1.42: SCULPTURE You are bending a strip of metal into an isosceles triang...
 4.4.7.42: VISUAL REASONING In the pattern below, each small triangle is an eq...
 4.4.8.42: VERIFYING CONGRUENCE Show that nABC and nDEF are right triangles an...
 4.4.4.42: Draw a figure that fits the description. (p. 42) A pentagon that is...
 4.4.5.42: Decide which method, SSS, SAS, or HL, can be used to prove that the...
 4.4.6.42: How many lines can be drawn that fit each description? Copy the dia...
 4.4.1.43: MULTIPLE CHOICE Which inequality describes the possible measures of...
 4.4.7.43: REASONING Let nPQR be an isosceles right triangle with hypotenuse }...
 4.4.8.43: MULTIPLE CHOICE A piece of paper is folded in half and some cuts ar...
 4.4.4.43: Draw a figure that fits the description. (p. 42) A quadrilateral th...
 4.4.5.43: Decide which method, SSS, SAS, or HL, can be used to prove that the...
 4.4.6.43: How many lines can be drawn that fit each description? Copy the dia...
 4.4.1.44: SLING CHAIRS The brace of a sling chair forms a triangle with the s...
 4.4.7.44: REASONING Explain how the Corollary to the Base Angles Theorem foll...
 4.4.8.44: CHALLENGE A triangle is rotated 908 counterclockwise and then trans...
 4.4.4.44: Write an equation of the line that passes through point P and is pe...
 4.4.6.44: The variable expressions represent the angle measures of a triangle...
 4.4.1.45: SLING CHAIRS The brace of a sling chair forms a triangle with the s...
 4.4.7.45: PROVING THEOREM 4.8 Write a proof of the Converse of the Base Angle...
 4.4.8.45: Simplify the expression. Variables a and b are positive. 2a 2 00 2 ...
 4.4.4.45: Write an equation of the line that passes through point P and is pe...
 4.4.6.45: The variable expressions represent the angle measures of a triangle...
 4.4.1.46: SLING CHAIRS The brace of a sling chair forms a triangle with the s...
 4.4.7.46: EXTENDED RESPONSE Sue is designing fabric purses that she will sell...
 4.4.8.46: Simplify the expression. Variables a and b are positive. (a 1 b) 2 ...
 4.4.4.46: Write an equation of the line that passes through point P and is pe...
 4.4.6.46: The variable expressions represent the angle measures of a triangle...
 4.4.1.47: SLING CHAIRS The brace of a sling chair forms a triangle with the s...
 4.4.7.47: REASONING FROM DIAGRAMS Use the information in the diagram to answe...
 4.4.8.47: Simplify the expression. Variables a and b are positive. 2a 1 2b2 (...
 4.4.4.47: Find the value of x. (p. 225) RTS708EG F4088x
 4.4.1.48: PROOF Prove the Corollary to the Triangle Sum Theorem on page 220.
 4.4.7.48: REASONING FROM DIAGRAMS Use the information in the diagram to answe...
 4.4.8.48: Simplify the expression. Variables a and b are positive. (p. 139) }...
 4.4.4.48: Find the value of x. (p. 225) L888 3583x8
 4.4.1.49: MULTISTEP PROBLEM The measures of the angles of a triangle are (2}...
 4.4.7.49: PROOF Write a proof. GIVEN c nABC is equilateral, CAD > ABE > BCF. ...
 4.4.8.49: Simplify the expression. Variables a and b are positive. (p. 139) }...
 4.4.1.50: PROVING THEOREM 4.2 Prove the Exterior Angle Theorem. (Hint: Find t...
 4.4.7.50: COORDINATE GEOMETRY The coordinates of two vertices of nTUV are T(0...
 4.4.8.50: Simplify the expression. Variables a and b are positive. (p. 139) }...
 4.4.1.51: EXTENDED RESPONSE The figure below shows an initial plan for a tria...
 4.4.8.51: Use the SSS Congruence Postulate to show nRST > nUVW. (p. 234) R(1,...
 4.4.1.52: ALGEBRA nABC is isosceles. AB 5 x and BC 5 2x 2 4. a. Find two poss...
 4.4.7.52: CHALLENGE The lengths of the sides of a triangle are 3t, 5t 2 12, a...
 4.4.1.53: CHALLENGE Use the diagram to write a proof of the Triangle Sum Theo...
 4.4.7.53: What quadrant contains the point? (p. 878) (22, 4)
 4.4.1.54: A and B are complementary. Find m A and m B. (p. 35) m A 5 (3x 1 16...
 4.4.7.54: What quadrant contains the point? (p. 878) (5, 22)
 4.4.1.55: A and B are complementary. Find m A and m B. (p. 35) m A 5 (4x 2 2)...
 4.4.7.55: Copy and complete the given function table. (p. 884) x 270 5y 5 x 2...
 4.4.1.56: A and B are complementary. Find m A and m B. (p. 35) m A 5 (3x 1 4)...
 4.4.7.56: Copy and complete the given function table. (p. 884) ? 220 1? 260 3
 4.4.1.57: Each figure is a regular polygon. Find the value of x. (p. 42) 12x ...
 4.4.7.57: Use the Distance Formula to decide whether }AB >}AC . (p. 15) A(0, ...
 4.4.1.58: Each figure is a regular polygon. Find the value of x. (p. 42) 3x 1...
 4.4.7.58: Use the Distance Formula to decide whether }AB >}AC . (p. 15) A(3, ...
 4.4.1.59: Each figure is a regular polygon. Find the value of x. (p. 42) x 1 ...
 4.4.7.59: Use the Distance Formula to decide whether }AB >}AC . (p. 15) A(3, ...
 4.4.1.60: Use the Symmetric Property of Congruence to complete the statement:...
 4.4.7.60: Use the Distance Formula to decide whether }AB >}AC . (p. 15) A(23,...
 4.4.1.61: Use the diagram at the right. (p. 124) 2143 If m 1 5 1278, find m 2...
 4.4.1.62: Use the diagram at the right. (p. 124) 2143 If m 4 5 1708, find m 1...
 4.4.1.63: Use the diagram at the right. (p. 124) 2143 If m 3 5 548, find m 1,...
Solutions for Chapter 4: Apply Triangle Sum Properties
Full solutions for Geometry (Holt McDougal Larson Geometry)  1st Edition
ISBN: 9780618595402
Solutions for Chapter 4: Apply Triangle Sum Properties
Get Full SolutionsGeometry (Holt McDougal Larson Geometry) was written by and is associated to the ISBN: 9780618595402. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4: Apply Triangle Sum Properties includes 416 full stepbystep solutions. Since 416 problems in chapter 4: Apply Triangle Sum Properties have been answered, more than 28932 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Geometry (Holt McDougal Larson Geometry), edition: 1.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Column space C (A) =
space of all combinations of the columns of A.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.