PROVING STATEMENTS ON TRIANGLE CONGRUENCE.pptx

PROVING STATEMENTS ON TRIANGLE CONGRUENCE

1. RECALLING TRIANGLE CONGRUENCE

2. Read each item carefully and choose the letter of the correct answer. What does “CPCTC” stand for? Congruent Parts of Corresponding Triangles are Congruent. Corresponding Portion of Congruent Triangles are Constant. Corresponding Parts of Congruent Triangles are Congruent. Congruent Portion of Corresponding Triangles are Constant.

3. Read each item carefully and choose the letter of the correct answer. What does “CPCTC” stand for? Congruent Parts of Corresponding Triangles are Congruent. Corresponding Portion of Congruent Triangles are Constant. Corresponding Parts of Congruent Triangles are Congruent. Congruent Portion of Corresponding Triangles are Constant.

4. 2. Which of the following statements is TRUE about CPCTC? If you prove that two triangles are congruent, then the corresponding sides and angles are congruent. Before you prove that two triangles are congruent, you must prove first that the corresponding sides and angles are congruent. CPCTC is used in proofs before it was proven that two triangles are congruent. After proving that two triangles are congruent, you cannot make conclusions that the remaining sides and angles are also congruent.

5. 2. Which of the following statements is TRUE about CPCTC? If you prove that two triangles are congruent, then the corresponding sides and angles are congruent. Before you prove that two triangles are congruent, you must prove first that the corresponding sides and angles are congruent. CPCTC is used in proofs before it was proven that two triangles are congruent. After proving that two triangles are congruent, you cannot make conclusions that the remaining sides and angles are also congruent.

6. Which of the following illustrates reflexive property of congruence?  

7. Which of the following illustrates reflexive property of congruence?  

8. What triangle postulate describes “If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.” SSS Postulate SAS Postulate ASA Postulate

9. What triangle postulate describes “If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.” SSS Postulate SAS Postulate ASA Postulate

10. What triangle postulate describes “If two angles and the included side of one triangle are congruent to the corresponding angles and included side of another triangle, then the triangles are congruent.” SSS Postulate SAS Postulate ASA Postulate

11. What triangle postulate describes “If two angles and the included side of one triangle are congruent to the corresponding angles and included side of another triangle, then the triangles are congruent.” SSS Postulate SAS Postulate ASA Postulate

12. Refer to Figure 1. If AB≅DE , BC≅EF and AC≅DF , then ∆ABC≅∆DEF by what postulate or theorem? A. AAS Congruence Theorem B. ASA Congruence Postulate C. SAS Congruence Postulate D. SSS Congruence Postulate

13. Refer to Figure 1. If AB≅DE , BC≅EF and AC≅DF , then ∆ABC≅∆DEF by what postulate or theorem? A. AAS Congruence Theorem B. ASA Congruence Postulate C. SAS Congruence Postulate D. SSS Congruence Postulate

14. Corresponding parts are marked in figure 2, how can you prove that ∆ABE≅∆DBC? A. AAS Congruence Theorem B. ASA Congruence Postulate C. SAS Congruence Postulate D. SSS Congruence Postulate

15. Corresponding parts are marked in figure 2, how can you prove that ∆ABE≅∆DBC? A. AAS Congruence Theorem B. ASA Congruence Postulate C. SAS Congruence Postulate D. SSS Congruence Postulate

16. PROVING STATEMENTS ON TRIANGLE CONGRUENCE HAZELYN C. ORLANDA Math 8 Teacher

17. From our previous lessons on triangle congruence, you learn that two triangles are congruent if the three pairs of corresponding sides have equal lengths, and the three pairs of corresponding angles have equal measures. However, applying SSS Postulate, SAS Postulate, and ASA Postulate, you can also prove that the two triangles are congruent, and when two triangles are congruent you can conclude that the remaining corresponding sides and angles are congruent since C orresponding P arts of C ongruent T riangles are C ongruent ( CPCTC ). To prove two segments or two angles are congruent, you must show that they are corresponding parts of congruent triangles. The following examples will help you prove statements on triangle congruence.

18. When two triangles are congruent, all six pairs of corresponding parts (angles and sides) are congruent. This statement is usually simplified as Corresponding Parts of Congruent Triangles are Congruent , or CPCTC . CPCTC

19. Prove ≅   Example Solution: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. ∠ A ≅ ∠D ≅   ∠ BCA ≅ ∠ECD ∆CAB ≅ ∆CDE ≅   Given Given Vertical angles are congruent Angle-Side-Angle (ASA) Postulate Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

20. Given: H is the midpoint of ≅ ∠I Prove: ≅   Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. H is the midpoint of   ≅   ∠ K ≅ ∠I ∆KGH ≅ ∆IGH ∠ GHK ≅ ∠JHI Given Given Definition of vertical angles ASA Postulate CPCTC ≅   ∠ GHK & ∠JHI are vertical angles Definition of midpoint Vertical Angles are congruent

21. Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Illustrative Examples: Given: AB ≅ ED , BC ≅ DC , AC ≅ EC Prove: ∠ BAC ≌ ∠ DEC B C A D E Proof:           Given Given Given SSS Postulate Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

22. Statements Reasons 1. 1. Given 2. 2. Given 3. 3. Given 4. △ABC ≌ △EDC 4. SSS Postulate 5. ∠BAC ≌ ∠DEC 5. Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Statements Reasons 1. Given 2. Given 3. Given 4. △ABC ≌ △EDC 4. SSS Postulate 5. ∠BAC ≌ ∠DEC 5. Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Illustrative Examples: Given: AB ≅ ED , BC ≅ DC , AC ≅ EC Prove: ∠ BAC ≌ ∠ DEC B C A D E Proof:

23. B C A D E

24. Perpendicular lines are lines, segments or rays that intersect to form right angles. The symbol ⊥ means is perpendicular to. In the figure, OB ⊥ RI. The right angle symbol in the figure indicates that the lines are perpendicular. R O I B

25. When learning about midpoints , it is also important to understand the concept of congruent segments. Congruent line segments are line segments with the same length. In a line segment, there is one point that will bisect the line segment into two congruent line segments. This point is called the midpoint .

26. R O I B If B is the midpoint of , then RB = IB.  

27. Analysis: You can prove ∠ BAC ≌ ∠ DEC if you already proved that △ ABC ≌ △ EDC. It is already given that . Marking the necessary sides shows that △ ABC ≌ △ EDC by SSS Postulate. Since the two triangles are congruent, ∠ BAC ≌ ∠ DEC by Corresponding Parts of Congruent Triangles are Congruent (CPCTC).  

28. Given: OB  RI , B is the midpoint of Prove: RO ≅ IO   Proof: R O I B Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. B is the midpoint of       ∠RBO and ∠IBO are right angles ∠RBO ≌ ∠IBO   △RBO ≌ △IBO   Given Definition of Midpoint Given Perpendicular lines form right angles All right angles are congruent Reflexive property SAS Postulate CPCTC

29. Given: OB  RI , B is the midpoint of RI Prove: RO ≅ IO Proof: R O I B Statements Reasons 1. B is the midpoint of 1. Given 2. 2. Definition of Midpoint 3. 3. Given 4. ∠RBO and ∠IBO are right angles 4. Perpendicular lines form right angles 5. ∠RBO ≌ ∠IBO 5. All right angles are congruent. 6. 6. Reflexive property 7. △RBO ≌ △IBO 7. SAS Postulate 8. 8. CPCTC Statements Reasons 1. Given 2. Definition of Midpoint 3. Given 4. ∠RBO and ∠IBO are right angles 4. Perpendicular lines form right angles 5. ∠RBO ≌ ∠IBO 5. All right angles are congruent. 6. Reflexive property 7. △RBO ≌ △IBO 7. SAS Postulate 8. CPCTC

30. Analysis: It is given that B is the midpoint of . As you know, midpoint divides segments into two congruent parts. Thus, by the definition of midpoint. It is also given that and perpendicular segments form a 90⁰ angle. Thus, ∠ RBO ≌ ∠ IBO because all right angles are congruent. By Reflexive Property, . This means that △ RBO ≌ △ IBO by SAS postulate and by Corresponding Parts of Congruent Triangles are Congruent.    

31. Given: ∠ OEL and ∠ OEV are right angles, ∠ EOL ≌ ∠ EOV Prove: ≅   Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. L O V E L O V E ∠EOL ≌ ∠EOV   ∠ OEL and ∠OEV are right angles OEL ≌ ∠OEV   OEL ≌ △OEV     Given Reflexive property Given All right angles are congruent ASA Postulate CPCTC

32. Given: ∠ OEL and ∠ OEV are right angles, ∠ EOL ≌ ∠ EOV Prove: EL ≅ EV Proof: Statements Reasons 1. ∠EOL ≌ ∠EOV 1. Given 2. 2. Reflexive Property 3. OEL and ∠OEV are right angles 3. Given 4. OEL ≌ ∠OEV 4. All right angles are congruent 5. OEL ≌ △OEV 5. ASA Postulate 6. 6. CPCTC Statements Reasons 1. ∠EOL ≌ ∠EOV 1. Given 2. Reflexive Property 3. Given 4. All right angles are congruent 5. ASA Postulate 6. CPCTC L O V E L O V E

33. Analysis: It is given that ∠ EOL ≌ ∠ EOV. The common side of both triangles can be proved as by the reflexive property of congruency. Also, it is already given that OEL and ∠ OEV are right angles. Thus, OEL ≌ ∠ OEV because all right angles are congruent. This means that OEL ≌ △ OEV by ASA Postulate and by Corresponding Parts of Congruent Triangles are Congruent.  

34. Definition of Angle Bisector An angle bisector is a line or ray that divides an angle into two congruent angles. ("Bisect" means to divide into two equal parts.) T S Y A If bisects and , then .  

35. L A E X Drill 1. A. Fill in the blanks with correct Statements and Reasons to complete the proof. The choices are written in the box below. Given: Prove: ∠ XAL ≌ ∠ LEX Proof:   Statements Reasons 1. 1. 2. 2. Given 3. 3. 4. 4. SSS Postulate 5. ∠XAL ≌ ∠LEX 5. Statements Reasons 1. 2. 2. Given 3. 4. 4. SSS Postulate 5. ∠XAL ≌ ∠LEX 5. CPCTC   Reflexive Property Given △ AXL ≌ △ ELX  

36. 2. Given: bisects ∠TSY and ∠TAY Prove: ∠STA ≌ ∠SYA Proof:   CPCTC Given ≌   ASA Postulate ∠ TAS ≌ ∠ YAS Definition of Angle Bisector T S Y A Statements Reasons 1. bisects ∠TSY and ∠TAY 1. Given 2. ∠TSA ≌ ∠YSA 2. 3. 3. Definition of Angle Bisector 4. 4. Reflexive Property 5. △TSA ≌ △YSA 5. 6. ∠STA ≌ ∠SYA 6. Statements Reasons 1. Given 2. ∠TSA ≌ ∠YSA 2. 3. 3. Definition of Angle Bisector 4. 4. Reflexive Property 5. △TSA ≌ △YSA 5. 6. ∠STA ≌ ∠SYA 6.

37. Drill 1. B. Fill in the blanks with correct Statements and Reasons to complete the two-column proof of each problem. 1. Given: BA ≌ YX , ∠ B ≌ ∠ Y, BC ≌ YZ Prove: AC ≌ XZ Proof: Statements Reasons ≌ 1. Given 2. 2. Given 3. ≌ 3. 4.  4. SAS Postulate 5. ≌ 5. Statements Reasons 1. Given 2. 2. Given 3. 4.  4. SAS Postulate 5. B A C Z X Y

38. 2. Given: XY ≌ XZ , WY ≌ WZ Prove: XU bisects ∠ YXZ Proof: Statements Reasons ≌ 1. 2. 2. Given 3. ≌ 3. 4. △XYW ≌ △XZW 4. 5. ∠YXW ≌ ∠ZXW 5. 6.  6. Definition of angle bisector Statements Reasons 1. 2. 2. Given 3. 4. △XYW ≌ △XZW 4. 5. ∠YXW ≌ ∠ZXW 5. 6.  6. Definition of angle bisector X W Y U Z

39. A. CPCTC B. Definition of Vertical Angles C. Vertical Angle Theorem D. Δ𝐴𝐺𝑁 ≅ Δ𝐸𝐺𝐿 Statements Reasons 1. ≅ 1. Given 2. ≅ 2. Given 3. ∠𝐴𝐺𝑁 𝑎𝑛𝑑 ∠𝐸𝐺𝐿 are vertical angles. 3. _____________________ 4. ∠𝐴𝐺𝑁≅∠𝐸𝐺𝐿 4. Vertical Angle Theorem 5. ____________________ 5. SAS Congruence Postulate 6. ≅ 6. ____________________ Statements Reasons 1. Given 2. Given 3. ∠𝐴𝐺𝑁 𝑎𝑛𝑑 ∠𝐸𝐺𝐿 are vertical angles. 3. _____________________ 4. ∠𝐴𝐺𝑁≅∠𝐸𝐺𝐿 4. Vertical Angle Theorem 5. ____________________ 5. SAS Congruence Postulate 6. ____________________ For items 1 to 3. Complete the proof by filling in the blanks with the correct statements and reasons. Options are found in the box. Given: GA ≅ GE ; GN ≅ GL Prove: AN ≅ EL Proof:

40. Drill 2. Complete each proof by stating the reason justifying each step. 1. Given: ≌ ≌ Prove: ∠CAR ≌ ∠REC Proof:   R A C E Statements Reasons 1. ≌ 1. 2. ≌ 2. 3. ≌ 3. 4. △CAR ≌ △REC 4. 5. ∠CAR ≌ ∠REC 5. Statements Reasons 1. 2. 3. 4. △CAR ≌ △REC 4. 5. ∠CAR ≌ ∠REC 5.