5-Minute Check on Lesson 6-35-Minute Check on Lesson 6-35-Minute Check on Lesson 6-35-Minute Check on Lesson 6-3 Transparency 6-4

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Determine if each pairs of triangles are similar. If so, write a similarity statement. Justify your statement.

1. 2. 3.

4. In the figure below, if RS // VT, then find y.Standardized Test Practice:

A CB D-0.8 0.8 1.2 4.8

∆BAC ~ ∆DECAA Similarity

∆GHI ~ ∆KLJSSS Similarity

B

No. Sides are notproportional

R

S

V

U

T

5

38 y

+ 12

A B

C

D E

9.0

6.75

4.8 7.63.65.7

K

L

J

G

H

I4.5

12

9

3.5

Lesson 6-4

Parallel Lines and Proportional Parts

Objectives

• Use proportional parts of triangle

• Divide a segment into parts

Vocabulary

• Midsegment: a segment whose endpoints are the midpoints of two sides of the triangle

From the Triangle Proportionality Theorem,

In ∆RST, RT // VU, SV = 3, VR = 8, and UT = 12. Find

SU.

S

Example 1a

Multiply.

Divide each side by 8.

Simplify.

Answer:

Answer: 15.75

In ∆ABC, AC // XY, AX=4, XB=10.5 and CY=6. Find BY.

B

Example 1b

In ∆DEF, DH=18, HE=36, and 2DG = GF. Determine

whether GH // FE. Explain.

In order to show that we

must show that

Since the sides have

proportional length.

Answer: since the segments have proportional

lengths,

Example 2a

In ∆WXZ, XY=15, YZ=25, WA=18 and AZ=32. Determine

whether WX // AY. Explain.

Answer: No; the segments are not in proportion since

X

Example 2b

In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in city blocks that the streets are apart. Find x.

Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem.

Triangle Proportionality Theorem

Multiply.

Divide each side by 13.

Answer: 32

Example 3

In the figure, Davis, Broad, and Main Streets are all parallel. The figure shows the distances in city blocks that the streets are apart. Find x.

Answer: 5

Example 3b

Find x and y.

To find x:

Given

Subtract 2x from each side.

Add 4 to each side.

To find y: The segments with lengths 5y and (8/3)y + 7 are congruent since parallel lines that cut off congruent segments on one transversal cut off congruent segments on every transversal.

Equal lengthsMultiply each side by 3 to eliminate the denominator.

Subtract 8y from each side.

Divide each side by 7.

Answer: x = 6; y = 3

Example 4a

Find a and b.

Answer: a = 11; b = 1.5

Example 4b

Summary & Homework

• Summary:– A segment that intersects two sides of a triangle

and is parallel to the third side divides the two intersected sides in proportion

– If two lines divide two segments in proportion, then the lines are parallel

• Homework: – Day 1: pg 311-2: 9,10, 14-18– Day 2: pg 312-3: 11, 12, 20, 21, 23-26, 33, 34