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Triangle Proportionality Theorem: Proof and Application

Learn about the Triangle Proportionality Theorem and its converse, including proofs, using similar triangles and information on proportional division of sides. Understand the theorem's statement, converse, and how it relates to parallel lines and triangles. Explore step-by-step proofs and application problems, with detailed explanations and calculations.

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Triangle Proportionality Theorem: Proof and Application

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  1. Proofs using Similar Triangles

  2. Information

  3. 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.

  4. 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

  5. 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

  6. Similar triangles

  7. 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

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