D. N. A. Are the following triangles similar? If yes, state the appropriate triangle similarity theorem. 9 2) 1) 15 12 8 3) Find the value of x and the length of PQ.

Parallel Lines and Proportional Parts Chapter 7-4

Use proportional parts of triangles. Divide a segment into parts. midsegment Standard 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. (Key) Lesson 4 MI/Vocab

Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. The converse is true also. B A E D C

Example #1 B A E D C 24 26 9.75 9

Find the Length of a Side Lesson 4 Ex1

Find the Length of a Side Substitute the known measures. Cross products Multiply. Divide each side by 8. Simplify. Lesson 4 Ex1

A. 2.29 B. 4.125 C. 12 D. 15.75 Lesson 4 CYP1

Find the value of x and y.

Determine Parallel Lines In order to show that we must show that Lesson 4 Ex2

Determine Parallel Lines Since the sides have proportional length. Lesson 4 Ex2

A. yes B. no C. cannot be determined A B C Lesson 4 CYP2

Midsegment Theorem The midsegment connecting the midpoints of two sides of the triangle is parallel to the third side and is half as long. C E B D A DE // AB and DE = AB

Midsegment of a Triangle Lesson 4 Ex3

Midsegment of a Triangle Use the Midpoint Formula to find the midpoints of Answer: D (0, 3), E (1, –1) Lesson 4 Ex3

Midsegment of a Triangle Lesson 4 Ex3

Midsegment of a Triangle If the slopes of slope of slope of Lesson 4 Ex3

Midsegment of a Triangle Lesson 4 Ex3

Midsegment of a Triangle First, use the Distance Formula to find BC and DE. Lesson 4 Ex3

Midsegment of a Triangle Lesson 4 Ex3

A. W (0, 1), Z (1, –3) B. W (0, 2), Z (2, –3) C. W (0, 3), Z (2, –3) D. W (0, 2), Z (1, –3) Lesson 4 CYP3

A. yes B. no A B Lesson 4 CYP3

A B A. yes B. no Lesson 4 CYP3

Parallel Proportionality Theorem B A F D C E If 3 // lines intersect two transversals, then they divide the transversals proportionally.

Example #2 Find ST SP // TQ // UR Corresponding Angle Thm. 9 U T S Q R 15 11 SP // TQ // UR Corresponding Angle Thm. Parallel Proportionality Theorem

Example #4 Solve for x and y 37.5 – x Solving for x J K M N L 7.5 9 13.5 x y 37.5 What is JL? 37.5 – x

Example #4 Solve for x and y Solving for y J K M N L 7.5 9 13.5 x y 37.5 JKL~JMN AA~Theorem

Proportional Segments MAPS 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. Lesson 4 Ex4

Proportional Segments Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem. Triangle Proportionality Theorem Cross products Multiply. Divide each side by 13. Answer: 32 Lesson 4 Ex4

In the figure, Davis, Broad, and Main Streets are all parallel 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. A. 4 B. 5 C. 6 D. 7 Lesson 4 CYP4

Subtract 2x from each side. Congruent Segments Find x and y. To find x: Given Subtract 2x from each side. Add 4 to each side. Lesson 4 Ex5

Congruent Segments To find y: The segments with lengths are congruent since parallel lines that cut off congruent segments on one transversal cut off congruent segments on every transversal. Lesson 4 Ex5

Multiply each side by 3 to eliminate the denominator. Congruent Segments Equal lengths Multiply each side by 3 to eliminate the denominator. Subtract 8y from each side. Divide each side by 7. Answer: x = 6; y = 3 Lesson 4 Ex5

Find a. A. B. 1 C. 11 D. 7 Lesson 4 CYP5

Find b. A. 0.5 B. 1.5 C. –6 D. 1 Lesson 4 CYP5

Homework Chapter 7-4 Pg 410 13-21, 26 – 27, 32 – 36, 61