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Name LESSON 4-6 Date Class Reteach Triangle Congruence: CPCTC Corresponding Parts of Congruent Triangles are Congruent (CPCTC) is useful in proofs. If you prove that two triangles are congruent, then you can use CPCTC as a justification for proving corresponding parts congruent. _ _ _ _ ! Given: AD CD , AB CB $ Prove: A C Proof: # !$#$ " 'IVEN N!"$N#"$ !"#" 'IVEN 333 !# #0#4# "$"$ 2EFLEX0ROPOF Complete each proof. _ , _ 1. Given: PNQ LNM, PN_ LN , N is the midpoint of QM . _ 0 - _ Prove: PQ LM Proof: . 1 0.1,.Given c. O01. 0.,. a. Given .is the mdpt. of -1. b. 1. Given Def. of midpt. O,-. d.CPCTC SAS -. 2. Given: UXW and UVW are right s. _ _ UX UV 5 Prove: X V 8 7 Proof: Statements Reasons 1. UXW and UVW are rt. s. 1. Given _ _ 2. UX UV _ 4. c. 2. a. Given 3. b. Reflex. Prop. of 4. d. HL 5. e. CPCTC _ 3. UW UW UXW UVW 5. X V Copyright © by Holt, Rinehart and Winston. All rights reserved. 01,- 46 6 Holt Geometry Name Date Class Reteach LESSON 4-6 Triangle Congruence: CPCTC continued You can also use CPCTC when triangles are on the coordinate plane. Given: C(2, 2), D(4, –2), E(0, –2), F(0, 1), G(–4, –1), H(–4, 3) Y ( 2 Prove: CED FHG # & X 0 2 Step 1 Plot the points on a coordinate plane. 2 ' $ % Step 2 Find the lengths of the sides of each triangle. Use the Distance Formula if necessary. d 2 2 x1) (y2 y1) (x2 CD (4 2) 2 (2 2)2 FG EC GH 4 [2 (2)] _ _ _ HF _ 2 2 [0 (4)] (1 3) 4 16 2 5 16 4 2 5 2 2 (2 0) (1 1) 4 16 2 5 DE 4 2 2 (4 0) _ = 16 4 2 5 _ So, CD FG , DE GH , and EC HF . Therefore CDE FGH by SSS, and CED FHG by CPCTC. Use the graph to prove each congruence statement. Y 2 3 Y " 9 3 2 # ! X X 1 2 0 2 0 2 * 2 2 7 3 , + 8 3. RSQ XYW 4. CAB LJK QR WX 13 , AB JK 5, BC KL RS XY 7, SQ YW 34 . 10 , CA LJ 53 . So So QRS WXY by SSS, and ABC JKL by SSS, and RSQ XYW by CPCTC. CAB LJK by CPCTC. 5. Use the given set of points to prove PMN VTU. M(–2, 4), N(1, –2), P(–3, –4), T(–4, 1), U(2, 4), V(4, 0) MN TU 3 5 , NP UV 2 5 , PM VT 65 . So MNP TUV by SSS, and PMN VTU by CPCTC. Copyright © by Holt, Rinehart and Winston. All rights reserved. 47 Holt Geometry Name LESSON 4-6 Date Class Name Practice A LESSON 4-6 Triangle Congruence: CPCTC Parts 1. CPCTC is an abbreviation of the phrase “Corresponding Triangles of Congruent Use the figure for Exercises _2 and _3. �B � �E, �C � �F, and AB � DE . � � � � � 3. Use CPCTC to name the other three pairs of congruent parts in the triangles. _ �D � _ _ AC DF � 4. Some hikers come to a river in the woods. They want to cross the river but decide to find out how wide it is first. So they set up congruent right triangles. The figure shows the river and the triangles. Find the width of the river, GH. BC � 5m � 3m � � Prove: �P � �R _ � � QS � QS 2. Reflexive Property of � 3. SAS 4. c. CPCTC Write a two-column proof. � XY � 3 BC � YZ � � 5 _ 4 SSS CPCTC 9. How do you know that �X � �A? Name LESSON 4-6 43 Date Class Holt Geometry 4. Reflex. Prop. of � 5. �FIH � �GHI 5. SAS 6. FH � GI 6. CPCTC 7. FH � GI 7. Def. of � segs. LESSON 4-6 � Prove: The diagonals of a parallelogram bisect each other. _ � 2. Given: FGHI is a rhombus. Prove: The diagonals of a rhombus are congruent, perpendicular, and bisect the vertex angles of the rhombus. _ _ _ � � Prove: �A � �C Proof: � ������� � ����� ����������� ������� ����� � ��� ������� ����� ������� � ����������������� � � Complete each proof. _ definition Possible _ _ answer: From the _IH is congruent to _ of a rhombus, FG, IF is congruent to GH, and IH is parallel to FG. By Alternate Interior Angles Theorem, �GFH is congruent to �IHF and �FGI is congruent_ by ASA. By CPCTC, FJ to �HIG. Therefore _ is congruent to �HIJ _ �FGJ _ is congruent to HJ and GJ is congruent to IJ. So �FJI is congruent to �GHJ by SSS. But �HIJ is also congruent to �FIJ by SSS. And so all four triangles are congruent by the Transitive_ Property of Congruence. By _ CPCTC and the Segment Addition Postulate, FH is congruent to GI. By �FJI, �GJF, �HJG, and �IJH CPCTC and the Linear_ Pair Theorem, _ By CPCTC, �GFH, � are right angles. So FH and GI are perpendicular. _ so FH bisects �IFG and �IHG. IFH, �GHF, and �IHF are congruent, _ Similar reasoning shows that GI bisects �FGH and �FIH. _ � _ 1. Given: �PNQ � �LNM, PN_ � LN , N is the midpoint of QM. _ � � _ Prove: PQ � LM Proof: � � ����������� Given ������� a. Given ��is the mdpt. of ��. Given 3. Rectangles, rhombuses, and squares are all types of parallelograms. Write a conjecture about the diagonals of a rectangle. b. �� c. ���� � �� � Prove: �X � �V � The diagonals of a square are congruent perpendicular bisectors that Statements Reasons 1. �UXW and �UVW are rt. �s. 1. Given 5. An isosceles trapezoid has one pair of noncongruent parallel sides, a pair of congruent nonparallel sides, and two pairs of congruent angles. What relationship do the diagonals of an isosceles trapezoid have? _ 2. UX � UV 2. a. Given 3. UW � UW 3. b. Reflex. Prop. of � _ 4. c. _ �UXW � �UVW 5. �X � �V The diagonals are congruent. Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 77 � � bisect the vertex angles of the square. _ ������� d.CPCTC Def. of midpt. Proof: 4. A square is a type of rhombus. Write a conjecture about the diagonals of a square. Copyright © by Holt, Rinehart and Winston. All rights reserved. � ���� SAS 2. Given: �UXW and �UVW are right �s. _ _ UX � UV The diagonals of a rectangle bisect each other. 45 Holt Geometry Triangle Congruence: CPCTC _ (Note: Be careful naming the triangles. The order of vertices matters.) _ Copyright © by Holt, Rinehart and Winston. All rights reserved. Class Reteach Given: AD � CD, AB � CB � Possible of a parallelogram, DC is congruent _ From the definition _ answer: _ to AB and DC is parallel to AB . By the Alternate Interior Angles Theorem, �BAC is congruent to �DCA and �CDB is congruent to _ �ABD. Therefore is congruent to �CDE _ by ASA. By CPCTC, DE _�ABE _ is congruent to BE and AE is congruent to CE. Congruent segments have equal lengths, so the diagonals bisect each other. � Date Corresponding Parts of Congruent Triangles are Congruent (CPCTC) is useful in proofs. If you prove that two triangles are congruent, then you can use CPCTC as a justification for proving corresponding parts congruent. � 1. Given: ABCD is a parallelogram. 44 Name Triangle Congruence: CPCTC � _ Copyright © by Holt, Rinehart and Winston. All rights reserved. Practice C Write paragraph proofs for Exercises 1 and 2. 3. Rt. � � Thm. _ 4. IH � IH _ 4 XZ � 8. Name the triangle congruence theorem that shows �ABC � �XYZ. Copyright © by Holt, Rinehart and Winston. All rights reserved. 2. FI � GH, �FIH and �GHI are right angles. 2. Def. of rectangle 3. �FIH � �GHI AC � 5 Reasons 1. Given _ _ � � � � Statements 1.FGHI is a rectangle. � � �� 7. Use the Distance Formula to find the length of each side. 3 � �� � � Prove: The diagonals of a rectangle have equal lengths. Possible answer: � 6. Plot these points: A (0, 0), B (0, 3), C(4, 0), X (�4, �3), Y(�4, 0), Z (0, �3). Draw segments to make �ABC and �XYZ. � 3. Given: FGHI is a rectangle. � Use the blank graph for Exercises 6–9. ����������� ��� ����������� ����� ������� ������������������ � AB � ����������� ����������������� ������� ����� �PQS � �RQS 4. �P � �R ��� ����� ������ Reasons 3. b. � � � 1. Given 2. a. � _ 2. Given: �L � �J, KJ � LM Prove: �LKM � �JMK _ � Statements _ � RQ,_ �PQS � �RQS 1. PQ _ � 15 ft 25 ft � � _ QS � QS, CPCTC, �PQS � �RQS 5. Given: PQ � RQ, �PQS � �RQS 15 ft corresponds with CE, so Heike could not have jumped this distance. Write a flowchart proof. 5m _ � 20 ft have jumped this distance. The distance along path CA is 25 ft because CA � � _ � distance along path BA is 20 ft because BA corresponds with DE, so Heike could Use the phrases in the word bank to complete this proof. _ Triangle Congruence: CPCTC same length as its corresponding side in �EDC. Heike could jump about 23 ft. The _ EF � 3m Practice B Vertical � Thm. the triangles are congruent by ASA, and each side in �ABC has the AAS 2. Name the triangle congruence theorem that shows �ABC � �DEF. �A Class 1. Heike Dreschler set the Woman’s World Junior Record for the long jump in 1983. She jumped about 23.4 feet. The diagram shows two triangles and a pond. Explain whether Heike could have jumped the pond along path BA or along path CA. Possible answer: Because �DCE � �BCA by the are Congruent.” � Date 4. d. HL 5. e. CPCTC 46 Holt Geometry Holt Geometry Name Date Class Name Reteach LESSON 4-6 LESSON 4-6 Triangle Congruence: CPCTC continued You can also use CPCTC when triangles are on the coordinate plane. Given: C(2, 2), D(4, –2), E(0, –2), F (0, 1), G(–4, –1), H(–4, 3) 2 Prove: �CED � �FHG � � � 0 �2 Step 1 Plot the points on a coordinate plane. 2 � � � Step 2 Find the lengths of the sides of each triangle. Use the Distance Formula if necessary. CD � �(4 � 2) 2 � (�2 � 2) � 2 FG � � (�1 � 1) � � �16 � 4 � 2 �5 GH � 4 DE � 4 EC � �� 2 2 �(�4 � 0) � � � �4 � 16 � 2 �5 � 2 2 �(2 � 0) _ _ � (1 � 3) � = �16 � 4 � 2 �5 ������� ���������� _ So, CD � FG, DE � GH, and EC � HF . Therefore �CDE � �FGH by SSS, and �CED � �FHG by CPCTC. � � 3 � 0 �2 2 � � � � 0 �2 ����� 2 2 � �2 � � 3. �RSQ � �XYW AB � JK � 5, BC � KL � � � So �QRS � �WXY by SSS, and �ABC � �JKL by SSS, and �RSQ � �XYW by CPCTC. �CAB � �LJK by CPCTC. M(–2, 4), N(1, –2), P(–3, –4), T(–4, 1), U(2, 4), V(4, 0) ������������ ��� ����� _ _ � � � a. �QNT is isosceles. So �MNP � �TUV by SSS, and �PMN � �VTU by CPCTC. 47 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name Date Holt Geometry Problem Solving LESSON 4-6 1. Two triangular plates are congruent. The area of one of the plates is 60 square inches. What is the area of the other plate? Explain. 2 triangles also have the same areas. _ _ 38 m � 3. A city planner sets up the triangles to find the distance RS across a river. Describe the steps that she can use to find RS. � 76 m � 38 m � ��������� ������������� � ����� _ ��������� ����� rt. �. PQ � RQ because 1. What are some reasons you would use an acronym? � PQ � RQ � 65 ft. �NQP � Possible answer: to abbreviate or make a statement simpler to understand or remember �SQR because vert. � are �. Therefore �NPQ � �SRQ by ASA. _ 2. What are some other acronyms you have used in your everyday life? By CPCTC, NP � SR. So SR � NP � 40 ft. Choose the best answer. Answers will vary. Students may mention FBI, IRS, RSVP, FAQ, or � others that are popular in text messaging. 4. A lighthouse and the range of its shining light are shown. What can you conclude? A x � y by CPCTC C �AED � �ADE by CPCTC B x � 2y D �AED � �ACB Examine the figure and answer the question. � 5. A rectangular piece of cloth 15 centimeters long is cut along a diagonal to form two triangles. One of the triangles has a side length of 9 centimeters. Which is a true statement? ���� � � ���� 6. Small sandwiches are cut in the shape of right triangles. The longest sides of all the sandwiches are 3 inches. One sandwich has a side length of 2 inches. Which is a true statement? A All the sandwiches have a side length of 2 inches by CPCTC. G The second triangle has a side length of 9 centimeters by CPCTC. B All the sandwiches are isosceles triangles with side lengths of 2 inches. H You cannot make a conclusion about the side length of the second triangle. C None of the other sandwiches have side lengths of 2 inches. J The triangles are not congruent. D You cannot make a conclusion using CPCTC. Copyright © by Holt, Rinehart and Winston. All rights reserved. � � F The second triangle has an angle measure of 15° by CPCTC. 49 ��������� ������������� �������� � ����� Holt Geometry Using an Acronym ����� � Class Reading Strategies � �P � �R because they are both Date One acronym used in geometry is CPCTC. Look at the breakdown of this acronym: 82 m 76 m by CPCTC. Therefore UV � XY � 82 m. c. �VMN is isosceles. An acronym is a word formed from the first letters of a phrase. For example, ASAP stands for “As Soon As Possible.” Acronyms can also combine the first letters or series of letters in a series of words, as in radar, which stands for radio detecting and ranging. 60 in ; Since the triangles are �, they have the same measures. So, the � 48 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name Triangle Congruence: CPCTC 82 m; �UVW � �XYW by SAS, so UV � XY b. �MPV is isosceles. 5. Given that m�NQT � 110°, find the measures of all the other angles between the rafters, beams, and struts of the queen-post truss. Label the angle measures directly on the figure. Class 2. An archaeologist draws the triangles to find the distance XY across a ravine. What is XY ? Explain. � 4. Using the information about the queen-post truss given above, prove each statement on a separate sheet of paper. Use any form of proof that you want. Proofs will vary. MN � TU � 3 � 5 , NP � UV � 2 �5 , PM � VT � �65 . Copyright © by Holt, Rinehart and Winston. All rights reserved. � ������������ �������� ���������������������� ��������� ������ �������� �������� ���������������� ������ ��� � ��� In the_ queen-post truss pictured at right, congruent queen posts NV ��� � ���� � ��� so that_ they are perpendicular to_ tie beam and _ TW are positioned _ _ _ _ ��� ��� PR and so that PV_ � VW_ � WR. Struts _ MV, SW, NT intersect � � _ _ and _ ��� ��� ��� ��� The the rafters so that PM � MN � NQ and RS � ST � TQ. ���� ��� ��� ���� _ � � � outer triangle, �PQR, is an isosceles triangle with base PR. ��� ��� ��� ��� 5. Use the given set of points to prove �PMN � �VTU. _ � 3. Given that m�KJZ � 33°, find the measures of all the other angles between the rafters, beams, and struts of the king-post truss. Label the angle measures directly on the figure. � �10, CA � LJ � �53 . So _ � ������ � ��� ��� ���� ��� ��� ���� ��� ��� ��� ��� � 4. �CAB � �LJK � RS � XY � 7, SQ � YW � � 34 . 4-6 � �������� Explanations may vary. � QR � WX � �13 , LESSON � 2. Explain why it must be true that �XJZ � �XZJ. (Hint: How is ZX related to �JKL?) �3 � ������ ���������������� �������� �������� ������� �������� � � � � ������ ����������� ����� Use the graph to prove each congruence statement. � Isosceles Triangles and Roof Trusses 1. Refer to the diagram of the king-post truss. Write a flowchart proof to show that �JKZ � �LKZ. � 2 2 �[0 � (�4)] � � _ _ _ HF � � [2 � (�2)] � � �4 � 16 � 2 �5 Challenge For example, a_ king-post truss is pictured at right. The king post, KZ, is a median of �JKL, and it provides support for the rafters. support for the rafters _ Additional _ comes from struts ZX and ZY , which are medians of �JKZ and �LKZ, respectively. The outer_ triangle, � �JKL, is an isosceles triangle with base JL. � 2 2 � x1) � (y2 � y1) �(x2� d� Class The wooden or metal framework that supports a roof is called a roof truss. The simplest type of roof truss has the shape of an isosceles triangle, as depicted by_ �ABC at right. In the diagram, _ the legs of the triangle, AB and _CB, represent sloping beams that are called rafters. The base, AC, represents the tie beam that “ties together” the rafters. However, large roofs require trusses with designs that are more complex than this. � � Date � � � � � _ _ 3. In this triangle, �C � �N and AC � LN. Assume that �ABC � �LMN. Name four other parts that are congruent using CPCTC. _ _ _ _ �B � �M ; �A � �L; CB � NM; AB � LM Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 78 50 Holt Geometry Holt Geometry