Triangle proportionality theorem Triangle proportionality theorem and converse: A line is parallel to the side of a triangle and intersects the two other sides if and only if it divides the sides proportionally. State the triangle proportionality theorem. A If XY is parallel to BC… X Y … then AX/XB = AY/YC. State the converse of thetriangle proportionality theorem. If AX/XB = AY/YC… B C … then XY is parallel to BC.

Triangle proportionality theorem proof Triangle proportionality theorem: If a line is parallel to the side of a triangle and intersects the two other sides, then it divides the sides proportionally. Prove the triangle proportionality theorem. given: XY∥ BC A corresponding angles postulate: ∠AXY ≅ ∠B, ∠AYX ≅ ∠C X Y △ABC ~ △AXY AA similarity postulate: ⇒AX/(AX+XB) = AY/(AY+YC) cross-multiply: AX·AY+AX·YC = AY·AX+AY·XB B C simplify: AX·YC = AY·XB divide by YC·XB: AX/XB = AY/YC

Converse of the triangle prop. theorem Converse of the triangle proportionality theorem: If a line divides two sides of a triangle proportionally, then it is parallel to the other side. Prove the converse of the triangle proportionality theorem. given: AX/XB=AY/YC AX·YC = AY·XB cross-multiply : A AX·YC+AX·AY= AY·XB+AX·AY add AX·AY: AX(AY+YC)= AY(AX+XB) factor: X Y AX(AC)= AY(AB) substitute: AX/AB = AY/AC rearrange: ∠A ≅ ∠A reflex. prop.: △ABC ~ △AXY SAS similarity postulate: B C ⇒∠AXY ≅ ∠B, ∠AYX ≅ ∠C conv. corr. ang. theorem: XY∥ BC

Summary problem In the triangle shown, find the values of x and y. A x+4 9 Since DE∥FG and BC∥FG, use the triangle proportionality theorem. 2x E D 12 y 7 G F C B First look at △AFG. Then look at △ABC. by the tri. prop. theorem: by the tri. prop. theorem: AD/DF = AE/EG AF/FB = AG/GC substitute known lengths and x : substitute known lengths: (x + 4)/2x= 9 /12 (x + 4 + 2x)/y = (9 + 12)/7 12x + 48 = 18x 28/y = 21/7 solving for x: x = 8 solving for x: y = 28 × 7 ÷ 21 = 8.05