Proportional Segments between Parallel Lines Look for the similar Triangles

Parallel Lines and Transversal Alternate interior angles are congruent if the lines are parallel Corresponding angles are congruent if the lines are parallel Could use SSI to prove lines parallel Think of the triangle inside of a triangle and how you might prove them similar Why are angles 1 and 2 congruent

Parallel/Proportionality Conjecture If a line parallel to one side of a triangle passes through the other two sides, then it divides the other two sides proportionally. (This ratio is not the scale factor between the triangles) The sides the parallel line intersects creates 2 segments on each side, those 4 segments are proportional Assume line is parallel to base of triangle

Converse If the segments are proportional then the lines are parallel. You need to know these conjectures forwards and backwards

Example You can use either the triangles or the proportional segments Using triangles – ab cancels

Example Find AB ___ if EC=9

Extended Parallel/Proportionality Conjecture If two or more lines pass through two sides of a triangle parallel to the third side, then they divided the two sides proportionally. This just means you can set up multiple proportions depending on the segments you are using on one side of the figure

What proportions can be set up

Example Show that sides are divided proportionally, set up similar triangles and just proportional sides.

Answers to Examples 1. 2. 3. Proportional segments x = 24 Here you need to set up similar triangles AB=27 X= 36

Homework Pg 627 1-13 odd