Given that , are the following proportions true?

21. In , is trisected

by and so that . Write two different proportions that follow from this information. ∠1 ⬵ ∠2 ⬵ ∠3 CE CD ∠ACB 䉭ABC

33. Use Theorem 5.6.1 and the drawing to complete the proof

of this theorem: “If a line is parallel to one side of a triangle and passes through the midpoint of a second side, then it will pass through the midpoint of the third side.” Given: with M the midpoint of ; Prove: N is the midpoint of RT Í MN 7 ST RS 䉭RST

28. In shown in Exercise 27, suppose that

, and are medians. Find the value of: a b

29. Given point D in the interior of , suppose that

, , , , and . Find RK.

30. Given point D in the interior of , suppose that

, , , and . Find

31. Complete the proof of this property:

If , then and PROOF Statements Reasons 1. 1. ? 2. 2. ? 3. 3. ? 4. 4. ? 5. 5. Means-Extremes Property symmetric form 6. 6. ?

32. Given:

, with , Prove: RX XS = ZT RZ Í YZ 7 RS Í XY 7 RT 䉭RST a + c b + d = c d a + c b + d = a b b a + c = ab + d ab + bc = ab + ad b c = a d a b = c d a + c b + d = c d a + c b + d = a b a b = c d KT KR . HT = 4 SH = 3 GS = 3 RG = 2 䉭RST KT = 3 HT = 5 SH = 4 GS = 4 RG = 3 䉭RST TH HS RK KT SK TG , RH 䉭RST

27. Given point D in the interior of

, which statements is are true? a b TK KR RG GS SH HT = 1 RK KT TH HS GS RG = 1 䉭RST

25. Given: bisects ,

, , , and Find: x HINT: You will need to apply the Quadratic Formula.

26. Given: bisects , , ,

, and Find: x MP = 3x - 1 RP = x + 1 NR = x MN = 2x ∠NMP MR VT = x + 2 RT = 2 - x SV = 3 RS = x - 6 ∠SRT RV

23. In right not shown with right

, bisects so that V lies on side . If , , and , find SV and VT.

24. Given: AC

is the geometric mean between AD and AB. , and Find: AC DB = 6 AD = 4 RT = 12 ST = 6 13 RS = 6 ST ∠SRT RV ∠S 䉭RST

22. In , ,

, and . With the angle bisectors as shown, which line segment is longer? a or ? b or ? c or ? FB AF DB CD EC AE m ∠ABC = 40° m ∠ACB = 60° m ∠CAB = 80° 䉭ABC A B E D C 1 2 3 B F A C E D D A B C V S T R 3 x – 6 x + 2 2 – x M P R 2 x x + 1 3 x – 1 x N T S R G K H D Exercises 27–30 Z S T R X Y S R M N T

34. Use Exercise 33 and the following drawing to complete

the proof of this theorem: “The length of the median of a trapezoid is one-half the sum of the lengths of the two bases.” Given: Trapezoid ABCD with median Prove: MN = 1 2 AB + CD MN

40. In , the altitudes of the triangle intersect at a point

in the interior of the triangle. The lengths of the sides of are , , and . a If , find RX and XS. HINT: Use the Pythagorean Theorem b If , find TY and YS. c If , find ZR and TZ. d Use results from parts a, b, and c to show that RX XS SY YT TZ ZR = 1. SZ = 168 13 RY = 168 15 TX = 12 TR = 13 ST = 15 RS = 14 䉭RST 䉭RST D A B M N X C Y Z W X 6 R M N S T 30° B F A C E D T S R X Z Y

39. In the figure, the angle bisectors of intersect at a

point in the interior of the triangle. If , , and , find: a CD and DB HINT: Use Theorem 5.6.3. b CE and EA c BF and FA d Use results from parts a, b, and c to show that BD DC CE EA AF FB = 1. CA = 4 BA = 6 BC = 5 䉭ABC

36. In right not shown with right

, bisects so that D lies on side . If and , find BD and AB. HINT: Let and . Then use the Pythagorean Theorem.

37. Given: not shown is isosceles with

; bisects and Find: BC

38. Given: with right

; and ; is trisected by and Find: TN , NM, and MR SN SM ∠RST ST = 6 m ∠R = 30° ∠RST 䉭RST AB = 1 ∠ABC BD m ∠ABC = m∠C = 72° 䉭ABC AB = 2x BD = x DC = 3 AC = 6 CB ∠BAC AD ∠C 䉭ABC

35. Use Theorem 5.6.3 to complete the proof of this theorem:

“If the bisector of an angle of a triangle also bisects the opposite side, then the triangle is an isosceles triangle.” Given: ; bisects ; Prove: is isosceles HINT: Use a proportion to show that YX = YZ. 䉭XYZ WX ⬵ WZ ∠XYZ YW 䉭XYZ