AMSCO'S Geometry. New York - Rye High School

366 Parallel Lines4. Triangle ABC is an isosceles right triangle with the right angle at C. Let P be the incenter ofABC.a. Find the measure of each acute angle of ABC.b. Find the measure of each angle of APB.c. Find the measure of each angle of BPC.d. Find the measure of each angle of CPA.e. Does the bisector of ACB also bisect APB? Explain your answer.5. Triangle ABC is an isosceles triangle with mC 140. Let P be the incenter of ABC.a. Find the measure of each acute angle of ABC.b. Find the measure of each angle of APB.c. Find the measure of each angle of BPC.d. Find the measure of each angle of CPA.e. Does the bisector of ACB also bisect APB? Explain your answer.6. In RST, the angle bisectors intersect at P. If mRTS = 50, mTPR = 120, andmRPS 115, find the measures of TRS, RST, and SPT.7. a. Draw a scalene triangle on a piece of paper or using geometry software. Label the triangleABC.b. Using compass and straightedge or geometry software, construct the angle bisectors ofthe angles of the triangle. Let AL be the bisector of A, BM be the bisector of B,and CN be the bisector of C, such that L, M, and N are points on the triangle.c. Label the incenter P.d. In ABC, does AP = BP = CP? Explain why or why not.e. If the incenter is equidistant from the vertices of DEF, what kind of a triangle isDEF?Applying Skills8. Prove Corollary 9.14a, “If a point is equidistant from the sides of an angle, then it lies on thebisector of the angle.”9. Given DB ' ABC and AD ' DC, when is ABDcongruent to DBC? Explain.D10. When we proved that the bisectors of the angles of a triangle intersect in a point, we beganby stating that two of the angle bisectors, AL and BM, intersect at P. To prove that theyintersect, show that they are not parallel. (Hint: AL and BM are cut by transversal AB.Show that a pair of interior angles on the same side of the transversal cannot besupplementary.)ABC