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Proofs using Similar TrianglesPowerPoint Presentation

Proofs using Similar Triangles

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### Proofs using Similar Triangles

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

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