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Proportions with Angle Bisectors

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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


When an angle within a triangle is bisected, the bisector divides the triangle proportionally

By definition, \overrightarrow{AC} divides \angle BAD equally, so \angle BAC \cong \angle CAD . The proportional relationship is \frac{BC}{CD}=\frac{AB}{AD} .

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 x .

Because the ray is the angle bisector it splits the opposite side in the same ratio as the sides. So, the proportion is:

\frac{9}{x} &= \frac{21}{14}\\	21x &= 126\\x &= 6

Example B

Determine the value of x that would make the proportion true.

You can set up this proportion just like the previous example.

\frac{5}{3} &= \frac{4x+1}{15}\\75 &= 3(4x+1)\\75 &= 12x+3\\72 &= 12x\\6 &= x

Example C

Find the missing variable:

Set up a proportion and solve like in the previous examples.

\frac{12}{4}&=\frac{x}{3}\\ 36&=4x\\ x&=9

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter7ProportionswithAngleBisectorsB


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. Set up a proportion and solve.

  \frac{20}{8}&=\frac{25}{y}\\ 20y&=200 \\ y&=10

2. Set up a proportion and solve.

 \frac{20}{y}&=\frac{15}{28-y}\\ 15y&=20(28-y)\\ 15y&=560-20y\\ 35y&=560\\ y&=16

3. Set up a proportion and solve.

 \frac{12}{z}&=\frac{15}{9-z}\\ 15z&=12(9-z)\\ 15z&=108=12z\\ 27z&=108\\ z&=4


Find the value of the missing variable(s).

Find the value of each variable in the pictures below.

Find the unknown lengths.

  1. Error Analysis

Casey attempts to solve for a in the diagram using the proportion \frac{5}{a}=\frac{6}{5} . What did Casey do wrong? Write the correct proportion and solve for a .

Solve for the unknown variable.

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