CONGRUENT TRIANGLES

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1 ONGRUN RINGLS wo triangles are congruent if there is a sequence of rigid transformations that carry one onto the other. wo triangles are also congruent if they are similar figures with a ratio of similarity of 1, that is 1 1. One way to prove triangles congruent is to prove they are similar first, and then prove that the ratio of similarity is 1. In these lessons, students find short cuts that enable them to prove triangles congruent in fewer steps, by developing five triangle congruence conjectures. hey are SSS, S, S, SS, and HL, illustrated below. SSS S S SS HL Note: S stands for side and stands for angle. HL is only used with right triangles. he H stands for hypotenuse and the L stands for leg. he pattern appears to be SS but this arrangement is NO one of our conjectures, since it is only true for right triangles. See the Math Notes boxes in Lessons and ore onnections Geometry

2 xample 2 Using the information in the diagrams below, decide if any triangles are congruent, similar but not congruent, or not similar. If you claim the triangles are congruent or similar, create a flow chart justifying your answer. a. b. X Z V W Y In part (a), Δ Δ by the SS conjecture. Note: If you only see S, observe that is congruent to itself. he roperty justifies stating that something is equal or congruent to itself. = = Δ Δ by SS prop. In part (b), ΔWXV ~ ΔZYV by the ~ conjecture. he triangles are not necessarily congruent; they could be congruent, but since we only have information about angles, we cannot conclude anything else. Vertical angles = ΔWXV ~ ΔZYV ~ Lines, alt. int. angles = here is more than one way to justify the answer to part (b). here is another pair of alternate interior angles ( WXV and ZYV) that are equal that we could have used rather than the vertical angles, or we could have used them along with the vertical angles. 78 ore onnections Geometry

3 In each diagram below, are any triangles congruent? If so, prove it. (Note: Justify some using flowcharts and some by writing two-column proofs.) F omplete a proof for each problem below in the style of your choice. 32. : R and MN bisect each other. 33. : bisects ; 1 2. rove: ΔN ΔMR rove: Δ Δ N R M :,, 35. : G SG, S rove: ΔF Δ rove: ΔG ΔSG F G S 36. : O M, O bisects MO 37. :, rove: ΔMO ΔO rove: Δ Δ M O arent Guide with xtra ractice 81

4 24. ΔS ~ ΔSJ by ~. J lt. int. angles = ΔS ~ ΔSJ ~ lt. int. angles = 25. ΔKRS ΔISR by HL ΔKRS & ΔISR are right triangles. KR = IS RS = RS ΔKRS ΔISR Refl. rop. HL 26. Yes 27. Yes right 's are Δ Δ S L vertical s are U Δ Δ S 28. Yes 29. Yes S V Δ Δ SS right 's are = 30. Not necessarily. 31. Yes ounterexample: J K I N R F Δ Δ Δ Δ F HL SSS F I H 32. N M and R by definition of bisector. N MR because vertical angles are equal. So, ΔN ΔMR by SS. 33. by definition of angle bisector. by reflexive so Δ Δ by S. 34. since alternate interior angles of parallel lines congruent so ΔF Δ by S. 84 ore onnections Geometry

5 35. G G by reflexive so ΔG ΔSG by SSS. 36. MO O because perpendicular lines form right angles MO O by angle bisector and O O by reflexive. So, ΔMO ΔO by S. 37. and since parallel lines give congruent alternate interior angles. by reflexive so Δ Δ by S. 38. by definition of bisector. since vertical angles are congruent. So Δ Δ by S. 39. RQ SQ since perpendicular lines form congruent right angles. Q Q by reflexive so ΔQR ΔQS by S. 40. SQ RQ by angle bisector and Q Q by reflexive, so ΔSQ ΔRQ by S. 41. KY HUG because parallel lines form congruent alternate exterior angles. Y + YU = YU + GU so Y GU by subtraction. G since perpendicular lines form congruent right angles. So ΔKY ΔHGU by S. herefore, K H since triangles have congruent parts. 42. MQL WLQ since parallel lines form congruent alternate interior angles. QL QL by reflexive so ΔMQL ΔWLQ by SS so WQL MLQ since congruent triangles have congruent parts. So ML WQ since congruent alternate interior angles are formed by parallel lines. 43. Yes 44. Not necessarily. 45. Not necessarily. Δ Δ SS arent Guide with xtra ractice 85