7.10: Proportions with Angle Bisectors
What if you were told that a ray was an angle bisector of a triangle? How would you use this fact to find unknown values regarding the triangle's side lengths? After completing this Concept, you'll be able to solve such problems.
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CK-12 Foundation: Chapter7ProportionswithAngleBisectorsA
James Sousa: Triangle Angle Bisector Theorem
James Sousa: Using the Triangle Angle Bisector Theorem to Determine Unknown Values
Guidance
When an angle within a triangle is bisected, the bisector divides the triangle proportionally
By definition, divides equally, so . The proportional relationship is .
Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides.
Example A
Find .
Because the ray is the angle bisector it splits the opposite side in the same ratio as the sides. So, the proportion is:
Example B
Determine the value of that would make the proportion true.
You can set up this proportion just like the previous example.
Example C
Find the missing variable:
Set up a proportion and solve like in the previous examples.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter7ProportionswithAngleBisectorsB
Vocabulary
Pairs of numbers are proportional if they are in the same ratio. An angle bisector is a ray that divides an angle into two congruent angles.
Guided Practice
Find the missing variables:
1.
2.
3.
Answers:
1. Set up a proportion and solve.
2. Set up a proportion and solve.
3. Set up a proportion and solve.
Practice
Find the value of the missing variable(s).
Find the value of each variable in the pictures below.
Find the unknown lengths.
- Error Analysis
Casey attempts to solve for a in the diagram using the proportion . What did Casey do wrong? Write the correct proportion and solve for .
Solve for the unknown variable.
Angle Bisector Theorem
The angle bisector theorem states that if a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.Proportion
A proportion is an equation that shows two equivalent ratios.Ratio
A ratio is a comparison of two quantities that can be written in fraction form, with a colon or with the word “to”.Image Attributions
Description
Learning Objectives
Here you'll learn how to set up and solve proportions with angle bisectors.