CONGRUENT TRIANGLES


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1 ONGUN INGL 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,,,, and HL, illustrated below. HL Note: 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 but this arrangement is NO one of our conjectures, since it is only true for right triangles. ee the Math Notes boxes in Lessons and arent Guide with xtra ractice 2014 M ducational rogram. ll rights reserved. 1
2 xample 1 Use your triangle congruence conjectures to decide whether or not each pair of triangles must be congruent. ase each decision on the markings, not on appearances. Justify each answer. a. b. c. d. e. f. In part (a), the triangles are congruent by the conjecture. he triangles are also congruent in part (b), this time by the conjecture. In part (c), the triangles are congruent by the conjecture. art (d) shows a pair of triangles that are not necessarily congruent. he first triangle displays an arrangement, while the second triangle displays an arrangement. he triangles could still be congruent, but based on the markings, we cannot conclude that they definitely are congruent. he triangles in part (e) are right triangles and the markings fit the HL conjecture. Lastly, in part (f), the triangles are congruent by the conjecture M ducational rogram. ll rights reserved. ore onnections Geometry
3 xample 2 Using the information given 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 Y In part (a), by the conjecture. Note: If you only see, observe that is congruent to itself. he eflexive roperty justifies stating that something is equal or congruent to itself. = m = m = eflexive rop. In part (b), XV ~ ZYV 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 = XV ~ ZYV ~ 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 ( XV 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. arent Guide with xtra ractice 2014 M ducational rogram. ll rights reserved. 3
4 roblems riefly explain if each of the following pairs of triangles are congruent or not. If so, state the triangle congruence conjecture that supports your conclusion. F G J 3. H I L V Y M N O U X F 7. H I L N G J M O 10. U X G H 3 4 I V 39º 112º J 112º 39º L F X Y Z Use your triangle congruence conjectures to decide whether or not each pair of triangles must be congruent. ase your decision on the markings, not on appearances. Justify your answer M ducational rogram. ll rights reserved. ore onnections Geometry
5 arent Guide with xtra ractice 2014 M ducational rogram. ll rights reserved Using the information given in each diagram 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 flowchart justifying your answer V X Y Z L O N V I L U N H I J
6 In each diagram below, are any triangles congruent? If so, prove it. (Note: Justify some using flowcharts and some by writing twocolumn proofs.) F omplete a proof for each problem below in the style of your choice. 32. : and MN bisect each other. 33. : bisects ; 1 2. rove: N M rove: N M :,, 35. : G G, rove: F rove: G G F G 36. : O M, O bisects MO 37. :, rove: MO O rove: M O M ducational rogram. ll rights reserved. ore onnections Geometry
7 38. : bisects, 39. :, rove: rove: 40. :, bisects 41. : U GY, Y HU, G, rove: HG G. rove: H Y U G H 42. : M L, M L rove: ML M L onsider the diagram at right. 43. Is? rove it! 44. Is? rove it! 45. Is? rove it! arent Guide with xtra ractice 2014 M ducational rogram. ll rights reserved. 7
8 nswers 1. F by. 2. GIH LJ by. 3. NM NO by. 4., so by HL. 5. he triangles are not necessarily congruent. 6. F by or. 7. GI GI, so GHI IJG by. 8. lternate interior angles = used twice, so LN NM by. 9. Vertical angles at 0, so O O by. 10. Vertical angles and/or alternate interior angles =, so UX VX by. 11. No, the length of each hypotenuse is different. 12. ythagorean heorem, so GH IHG by. 13. um of angles of triangle = 180º, but since the equal angles do not correspond, the triangles are not congruent. 14. F + F = F +, so F by. 15. XZ XZ, so XZ YXZ by. 16. by 17. by, with by the eflexive roperty. 18. VX ZXY by, with VX ZXY because vertical angles are. 19. by, with by the eflexive roperty. 20. L L by HL, with L L by the eflexive roperty. 21. Not necessarily congruent. 22. V ~ IV by ~ qual markings V ~ IV Vert. angles = ~ 23. LUN and H are not necessarily similar based on the markings M ducational rogram. ll rights reserved. ore onnections Geometry
9 24. ~ J by ~ J given lt. int. angles = ~ J ~ lt. int. angles = 25. I by HL & I are right triangles. HL I = I = efl. rop. 26. Yes 27. Yes ight s are Vertical s are eflexive rop. Δ Δ Δ Δ 28. Yes 29. Yes ight s are eflexive rop. Δ Δ eflexive rop. Δ Δ 30. Not necessarily. 31. Yes ounterexample: Δ ΔF HL arent Guide with xtra ractice 2014 M ducational rogram. ll rights reserved. 9
10 32. N M and by definition of bisector. N M because vertical angles are equal. o, N M by. 33. by definition of angle bisector. by reflexive so by. 34. since alternate interior angles of parallel lines congruent so F by. 35. G G by reflexive so G G by. 36. MO O because perpendicular lines form right angles MO O by angle bisector and O O by reflexive. o, MO O by. 37. and since parallel lines give congruent alternate interior angles. by reflexive so by. 38. by definition of bisector. since vertical angles are congruent. o by. 39. since perpendicular lines form congruent right angles. (eflexive rop.) so by. 40. by angle bisector and by reflexive, so by. 41. Y 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. o Y HGU by. herefore, H since triangles have congruent parts. 42. ML L since parallel lines form congruent alternate interior angles. L L by the eflexive roperty so ML L by, then L ML since congruent triangles have congruent parts. o ML since congruent alternate interior angles are formed by parallel lines. 43. Yes lt. Int. s are eflexive rop. 44. Not necessarily. 45. Not necessarily. Δ Δ M ducational rogram. ll rights reserved. ore onnections Geometry
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