Proportional parts
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Proportional Parts. Lesson 5-4. F. C. A. B. D. E. Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.. Similar Polygons. C. B. 1. 2. D. A. 4. 3. E. Triangle Proportionality Theorem.

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Proportional Parts

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Proportional parts

Proportional Parts

Lesson 5-4

Lesson 5-4: Proportional Parts


Proportional parts

F

C

A

B

D

E

Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.

Similar Polygons

Lesson 5-4: Proportional Parts


Proportional parts

C

B

1

2

D

A

4

3

E

Triangle Proportionality Theorem

If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional length.

Converse:

If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side.

Lesson 5-4: Proportional Parts


Proportional parts

B

D

E

6

9

C

A

x

4

B

4x + 3

2x + 3

D

E

9

5

A

C

Examples………

Example 1:

If BE = 6, EA = 4, and BD = 9, find DC.

6x = 36 x = 6

Example 2:

Solve for x.

Lesson 5-4: Proportional Parts


Theorem

Theorem

A segment that joins the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side.

R

M

L

T

S

Lesson 5-4: Proportional Parts


If two triangles are similar

If two triangles are similar:

(1) then the perimeters are proportional to the measures of the corresponding sides.

Lesson 5-4: Proportional Parts


Proportional parts

E

B

F

D

C

A

25

15

4

20

Example:

Given: ΔABC ~ ΔDEF, AB = 15, AC = 20, BC = 25, and DF = 4. Find the perimeter of ΔDEF.

The perimeter of ΔABC is 15 + 20 + 25 = 60.

Side DF corresponds to side AC, so we can set up a proportion as:

Lesson 5-4: Proportional Parts


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