1. Entry Task  Find the value of x in each figure  1. 2. 10 x 4 x 6 14

2. Proportions in Triangles Lesson 7-5

3. Investigation 1 Parallels and Proportionality  For this investigation you will need a ruler and a protractor.  Step 1: In the figure below, find the value of x. 8 12 16 x A D B E C 8 12 D E C 8 16 x A D B 24 x + 12 A D B

5. Now let’s see if the converse is true?  Complete steps 3-9 of Investigation 1. When you are finished, use your observations to write a conjecture about the ratios of the lengths of the segments that are cut off by parallel lines.  Our book calls this theorem the Side-Splitter Theorem.  If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.  The converse of this theorem is also true:  If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

6. Corollary to the Side-Splitter Theorem  This extends the theorem to multiple parallel lines.  If three parallel lines intersect on two transversals, then the segments intercepted on the transversal are proportional. 

7. Investigation 2  Complete this investigation and compare your results with others near you.  Use your observations to write a conjecture.

8. Triangle-Angle-Bisector Theorem  The angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

9. Lesson Check  Find the value of x in each figure. 1. 2. 3. Find the value of x and y in the figure.

10.  One more investigation… Investigating corresponding parts of triangles.  In this investigation, you will look at the relationship between corresponding altitudes, corresponding medians, and corresponding angle bisectors of similar triangles.  Proportional Parts Conjecture:  If two triangles are similar, then the corresponding altitudes, medians, and angle bisectors are proportional to the corresponding sides.

11. Homework Page 475 (10-22 evens, 24-34 all, 36-44 evens)