Section 8.6 Proportions and Similar Triangles OBJECTIVE: To use the Triangle Proportionality Theorem and the Triangle-Angle-Bisector Theorem BIG IDEAS: Visualization and Proportionality ESSENTIAL UNDERSTANDINGS: When two or more parallel lines intersect other lines, proportional segments are formed The bisector of an angle of a triangle divides the opposite side into two segments with lengths proportional to the sides of the triangle that form the angle MATHEMATICAL PRACTICE: Construct viable arguments and critique the reasoning of others

Proportionality Theorems Triangle Proportionality Theorem: If a line is ____________________ to one side of a triangle and ____________________ the other two sides, then it divides those sides _________________________ Converse of the Triangle Proportionality Theorem: If a line ____________________ two sides of a triangle _________________________, then it is ____________________ to the third side

Proportionality Theorems Corollary to the Triangle Proportionality Theorem: If _______________ parallel lines ____________________ two transversals, then the ____________________ intercepted on the transversals are _________________________ Triangle-Angle-Bisector Theorem: If a _______________ bisects an angle of a triangle, then it divides the ____________________ side into two segments that are _________________________ to the other two sides of the triangle

Examples 1. In the diagram, What is the length of ?

Examples 2. In the diagram, What is the lengths of ?

Examples 3. In the diagram, What is the length of ?

Examples 4. In the diagram, . Find

Examples 5. Find the value of x