Mrs. McConaughy Geometry 1

LESSON 8.5 Proportions and Similar Triangles

OBJECTIVES: To use proportionality theorems to calculate segment lengthsTo solve real-life problems by using proportions in triangles

Mrs. McConaughy Geometry 2

THEOREM: TRIANGLE PROPORTIONALITY THEOREM If a line parallel to one side of a triangle, intersect the othertwo sides, then _________________________________________.

it divides the other two sides proportionally

PRT RU

TU QS, then = TQ US.

I f

Mrs. McConaughy Geometry 3

THEOREM: CONVERSE OF TRIANGLE PROPORTIONALITY THEOREM If a line divides two sides of a triangle proportionally, then___________________________________________________________

PRT RU

= , then TU QSTQ US.

I f

it is parallel to the third side.

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EXAMPLE: Finding the length of a segment

Use the TriangleProportionalityTheorem to find y.

CM = ___ MB ___ = ___

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EXAMPLE: Determining Parallel Lines

Given the diagram, determine whether MN || GH.

LM = ___ = ___

MGLN = ___ = ___

NH

MN ___________ GH

because________.

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THEOREM: COROLLARY TO TRIANGLE PROPORTIONALITY THEOREM If three parallel lines intersect two transversals, then __________________________________________. a = __

b

they divide the sides proportionally.

Mrs. McConaughy Geometry 7

EXAMPLE: Using the corollary to the Triangle Proportionality Theorem The segments joining the sides of trapezoid RSTU are parallel to its bases. Find x and y.

Mrs. McConaughy Geometry 8

THEOREM: TRIANGLE-ANGLE- BISECTOR THEOREM If a ray bisects an angle of a triangle, then it __________________________________________________________________________________________.

divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

a = ___; a = ___b p

a = __; a = __b p

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EXAMPLE: Using the Triangle-Angle-Bisector Theorem

Find the value of x.IG = ___

GH ___ = ___

Mrs. McConaughy Geometry 10

FINAL CHECKS FOR UNDERSTANDING

1. In the diagram, 1 2 3. PQ = 9, QR = 15, and ST = 11. What is the length of ?

TU

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2. Find x.

3. The real estate term for distance along the edge of a piece of property that touches the ocean is “ocean frontage.” Find the ocean frontage for each lot shown.

Which of these lots should be listed for the highest price?

Mrs. McConaughy Geometry 12

Homework Assignment: