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Chapter 4 gal lour! Proofs with * may be key stepped. All others should be done in a full 2 column fashion. 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. Identify each line (not part of ABC ) as an altitude, a median and a perpendicular bisector. A Draw all three altitudes. B C E 2 1 2. A S 5 3 T 4 B Solve if you have enough information. If 1 3 , then which segments must be congruent? ______________________ If ES ET , m2 75, m5 3x , then x = ________ If 2, 5 supplements. ES = 3y+5, and ET = 25 – y, then y = __________ L B F M J K D 3. If M is on the perpendicular bisector of JL , then M is equidistant from ___________ and ____________ If M is on the perpendicular bisector JK , then ________ = ____________ 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 P *5. Given: HNI is isosceles with base HI ; LI and HK are medians of HPI ; NJ is altitude of HNI Prove: LM MK - Key step here. M K L N H J I N *6. Given: PN // MQ , MS PR P R PN MQ Prove: NMS QPR S M Q L P O Q K N a. LN is a median. KNL MNL ___________ b. LN is an altitude. KNL MNL ___________ c. LN is a perpendicular bisector. KNL MNL ___________ M 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. d. LN is angle bisector of KLM ; KLM is an isosceles with base KM . KNL MNL ___________ e. KO and PM are medians of isosceles KLM . PQK OQM __________ A 1 Given: AB // DC ; AD // BC B 2 Prove: AE CE; DE BE E 4 D *8. 3 C Given: 1 2; AE BD Prove: 5 6 A 1 2 3 4 B C D 5 6 E 9. 10. U N In the figure to the left, PU = PT, U , T are right angles, and mPNU 5(mNPT ) and PUN PTN Find mNPT . P T T S 11. 12. M Given: MTO is equilateral; TN bisects MTO . Which of the following cannot be proved with the given information? a. S and P are midpoints. b. TN MO c. M O MTO d. N is the midpoint of MO P N O With congruences shown, Solve for x. x 145 Given the figure to the left, which one of the following can be used to prove the triangles congruent? x x 13. L M Q P R 14. O N a. AAS only b. HL only c. AAS or SAS d. Not congruent In the picture, LPM NPO ; Name three other pairs of triangles that can be proved congruent.