By Kaufman R. M.

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**Sample text**

We have drawn the median AD, so BD ϭ DC. We need to prove that ∠ BAD ϭ ∠ CAD. Proof In ∆ABD and ∆ACD, AB ϭ AC (given) BD ϭ CD (median) AD ϭ AD (common) Then ∆ABD ϵ ∆ACD (side-side-side) Therefore ∠BAD ϭ ∠CAD (triangles congruent) Then the median bisects the vertical angle. A B D C Notice that the same pair of congruent triangles show that ∠B ϭ ∠C, the Isosceles Triangle Property. Had we used this instead of the common side AD, the triangle congruence combination would have been side-angle-side.

ABD:∆ADC b. ∆ABD:∆ABC c. ∆BFD:∆BFA d. ∆BFD:∆BDA e. ∆CDF:∆CFA f. ∆CFD:∆ABC A 2 F 1 B 56 CHAPTER 2 1 D 3 C 2. Calculate the unknown in each of the following. a. b. a c. 5 3. Find the lengths of a and b in the given triangle. 4 3 5 a b 4 Application T PT ᎏ. ϭ ᎏ21ᎏ, determine ᎏ 4. If ᎏPTᎏ S SQ P T U S R Q Part B 5. If BD ϭ DC and AF ϭ FD, prove that ∆ABF 1 ᎏ ϭ ᎏᎏ a. ᎏ ∆ABC 4 c. Prove that ∆ABF = ∆AFC. 6. If ᎏabᎏ ϭ ᎏdcᎏ, prove a–b c–d ᎏ ϭ ᎏᎏ a. ᎏ b d Thinking/Inquiry/ Problem Solving A ∆AFC 1 ᎏ ϭ ᎏᎏ b.

B D C Part B P 6. In the given diagram, PQ ϭ PR and PS ϭ PT. Prove that ∠QRS ϭ ∠RQT. Q S 32 CHAPTER 2 R T Application e t chnology APPENDIX P. 502 7. In quadrilateral ABCD, the diagonal AC bisects ∠DAB and AB ϭ AD. Prove that AC bisects ∠BCD and BC ϭ DC. B x x A C D 8. In the given diagram, ABE and CBD are straight lines. If AB ϭ BE, AC and AB are perpendicular, and DE and BE are also perpendicular, prove that AC ϭ DE. D B A E C 9. In ∆PQR, PQ ϭ PR and ∠QPT ϭ ∠RPT. Prove that ∠SQT ϭ ∠SRT. P S Q T R 10.