8. Suppose that you wish to prove that SAS

30. Given: bisects

and are right angles Prove: 䉭PQR ⬵ 䉭NMR ∠Q ∠M MQ PN P Q M N R R S T V Exercises 31, 32

31. Given: and

Prove: 䉭RST ⬵ 䉭SRV RV ⬵ TS ∠VRS ⬵ ∠TSR

37. In quadrilateral ABCD, and are perpendicular

bisectors of each other. Name all triangles that are congruent to: a b c 䉭ABD 䉭ABC 䉭ABE BD AC

41. Given: ;

is the perpendicular bisector of ; is the perpendicular bisector of . Prove: AR ⬵ RC BC RT AB RS ∠ABC

40. Given: and

Prove: and 䉭TPQ ⬵ 䉭VQP 䉭SPV ⬵ 䉭SQT ST ⬵ SV SP ⬵ SQ In Exercises 39 and 40, complete each proof.

39. Given: Plane M

C is the midpoint of and Prove: 䉭ABC ⬵ 䉭DEC AB 7 ED AD ⬜ BE EB

38. In and

, you know that , , and . Before concluding that the triangles are congruent by ASA, you need to show that . State the postulate or theorem that allows you to confirm this statement . ∠B ⬵ ∠E ∠B ⬵ ∠E AB ⬵ DE ∠C ⬵ ∠F ∠A ⬵ ∠D 䉭DEF 䉭ABC

36. Given that and

, does it follow that ? Cite the method, if any, used in reaching this conclusion. 䉭ABC ⬵ 䉭EDC BC ⬵ DC ∠A ⬵ ∠E

34. Given that and

, does it follow that ? Which method, if any, did you use?

35. Given that and

, does it follow that ? If so, cite the method used in arriving at this conclusion. 䉭ABC ⬵ 䉭EDC ∠B ⬵ ∠D ∠A ⬵ ∠E 䉭RST ⬵ 䉭RVU ST ⬵ UV ∠S ⬵ ∠V

32. Given: and

Prove: In Exercises 33 to 36, the methods to be used are SSS, SAS, ASA, and AAS.

33. Given that , does it follow that

is also congruent to ? Name the method, if any, used in arriving at this conclusion. 䉭RVT 䉭RSU 䉭RST ⬵ 䉭RVU 䉭RST ⬵ 䉭SRV ∠TRS ⬵ ∠VSR VS ⬵ TR S V U T R Exercises 33, 34 A B C D E Exercises 35, 36 A B D C E A B C D E F E M B D A C P S Q T V B T C S A R Recall that the definition of congruent triangles states that all six parts three sides and three angles of one triangle are congruent respectively to the six corresponding parts of the second triangle. If we have proved that by SAS the congru- ent parts are marked in Figure 3.14, then we can draw conclusions such as and . The following reason CPCTC is often cited for drawing such conclu- sions and is based on the definition of congruent triangles. AC ⬵ DF ∠C ⬵ ∠F 䉭ABC ⬵ 䉭DEF CPCTC Hypotenuse and Legs of a Right Triangle HL Pythagorean Theorem Square Roots Property Corresponding Parts of Congruent Triangles 3.2 KEY CONCEPTS A B C a Figure 3.14 D E F b General Rule: In a proof, two triangles must be proven congruent before CPCTC can be used to verify that another pair of sides or angles of these triangles are also congruent. Illustration: In the proof of Example 1, statement 5 triangles congruent must be stated before we conclude that by CPCTC. TZ ⬵ VZ STRATEGY FOR PROOF 왘 Using CPCTC EXAMPLE 1 GIVEN: bisects See Figure 3.15. PROVE: TZ ⬵ VZ WT ⬵ WV ∠TWV WZ T V Z W Figure 3.15 PROOF Statements Reasons 1. bisects 2. 3. 4. 5. 6. TZ ⬵ VZ 䉭TWZ ⬵ 䉭VWZ WZ ⬵ WZ WT ⬵ WV ∠TWZ ⬵ ∠VWZ ∠TWV WZ 1. Given 2. The bisector of an angle separates it into two 3. Given 4. Identity 5. SAS 6. CPCTC ⬵ ∠s 쮿 Exs. 1–3 CPCTC: Corresponding parts of congruent triangles are congruent.