Solved Examples: Applying the Triangle Angle Bisector Theorem with Diagrams

📚 Quick Study Guide

• 📐 Theorem Definition: The Triangle Angle Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the other two sides.
• ✏️ Formula: If $\overline{AD}$ bisects $\angle BAC$ in $\triangle ABC$, then $\frac{BD}{DC} = \frac{AB}{AC}$.
• 📏 Application: This theorem is useful for finding unknown side lengths in triangles when you know an angle bisector.
• 💡 Key Idea: The ratio of the two segments created by the angle bisector on one side of the triangle is equal to the ratio of the other two sides of the triangle.
• ✍️ Example: If $AB = 6$, $AC = 8$, and $BD = 3$, then using the theorem, $\frac{3}{DC} = \frac{6}{8}$. Solving for $DC$, we get $DC = 4$.

Practice Quiz

1. In $\triangle ABC$, $\overline{AD}$ bisects $\angle BAC$. If $AB = 10$, $AC = 15$, and $BD = 4$, find $DC$.

   

   1. 6
   2. 8
   3. 10
   4. 12

2. In $\triangle PQR$, $\overline{QS}$ bisects $\angle PQR$. If $PQ = 8$, $QR = 12$, and $PS = 6$, find $SR$.

   

   1. 7
   2. 9
   3. 10
   4. 11

3. In $\triangle XYZ$, $\overline{YW}$ bisects $\angle XYZ$. If $XY = 9$, $YZ = 12$, and $XW = 6$, find $WZ$.

   

   1. 6
   2. 8
   3. 9
   4. 10

4. In $\triangle DEF$, $\overline{EG}$ bisects $\angle DEF$. If $DE = 14$, $EF = 21$, and $DG = 8$, find $GF$.

   

   1. 10
   2. 12
   3. 14
   4. 16

5. In $\triangle ABC$, $\overline{BD}$ is an angle bisector. If $AB = 5$, $BC = 8$ and $AD = 3$, find $DC$.

   

   1. 4.8
   2. 5.2
   3. 6
   4. 7

6. In $\triangle LMN$, $\overline{NP}$ bisects $\angle LNM$. If $LN = 7$, $NM = 9$ and $LP = 4$, find $PM$.

   

   1. 4.14
   2. 5.14
   3. 6
   4. 7

7. In $\triangle QRS$, $\overline{ST}$ bisects $\angle QSR$. If $QS = 11$, $SR = 13$ and $QT = 5$, find $TR$.

   

   1. 5.91
   2. 6.91
   3. 7
   4. 8