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Chapter 4 gal lour! All work and proofs must be done on a separate piece of paper to receive credit. Proofs with * may be key stepped. All others should be done in a full 2 column fashion. As always, the standard is 100%. Know all of your previous theorems, postulates and definitions as well as the new ones from chapter 4: Theorems: SSS, SAS, AAS, ASA, HL Isosceles Triangle Theorem (If 2 sides congruent, opp angles congruent) Iso Triangle con: If 2 angles congruent, opp sides congruent Equilateral triangle has three 60 degree angles Angle bisector of vertex angle on isosceles triangle is also a perpendicular bisector to the opposite side A point lies on the perpendicular bisector of a segment, if and only if the point is equidistant from the endpoints of the segment. A point lies on the bisector of an angle if and only if it is equidistant from the sides of the angle. Vocabulary: Median, Altitude, Perpendicular bisector, Isosceles 1 a. Identify each line (not part of ABC ) as an altitude, a median and a perpendicular bisector b. Draw all three altitudes. A B C E 2 1 2. A 5 3 4 T S B Solve If 1 3 , then which segments must be congruent? ____________________ a. b. Let ES ET , m2 75, m5 3x , then x = ________ c. Given: 2 and 5 supplements, 1 4 . AE = 2x + 3y - 1, ES = 2x + 2y, ET = 5x – y +3 and EB = -3x + 5y + 6. x = _______________, y = _________________ L B F M J K D 3. a. If M is on the perpendicular bisector of JL , then M is equidistant from ___________ and ____________ b. If M is on the perpendicular bisector JK , then ________ = ____________ c. If M is equidistant from K and L, then M is on the _________________ E D G 4. Given: AD, BE bisect each other at C Prove: GC FC C A F B L M Q P R O N 5. In the picture, LPM NPO ; Name three other pairs of triangles that can be proved congruent. N *6. Given: PN // MQ , MS PR P R PN MQ Prove: NMS QPR S M Q 7. Identify the theorem or postulate that can be used, if any, to prove the following triangles congruent. If you cannot prove the triangles congruent, put an “X” on the line. L P O Q K N M KNL MNL a. LN is a median. ___________ d. LN is angle bisector of KLM ; KLM is an isosceles with base KM . KNL MNL ___________ KNL MNL b. LN is an altitude. ___________ e. KO and PM are medians of isosceles KLM . PQK OQM __________ c. LN is a perpendicular bisector. KNL MNL ___________ A 1 8. Given: AB // DC ; AD // BC B 2 Prove: AE CE; DE BE E 4 D 3 C U N P T 9. In the figure to the left, PU = PT, U , T are right angles, and mPNU 5(mNPT ) and PUN PTN Find mNPT . 10. Given: MTO is equilateral; TN bisects MTO . Which of the following cannot be proved with the given information? T S a. S and P are midpoints. b. TN MO c. M O MTO d. N is the midpoint of MO P M N O 11. With congruences shown, Solve for x. x 145 12. Given the figure to the left, which one of the following can be used to prove the triangles congruent? x x a. AAS only b. HL only c. AAS or SAS d. Not congruent