What if the cities of Verticville, Triopolis, and Angletown were joining their city budgets together to build a centrally located airport? There are freeways between the three cities and they want to have the freeway on the interior of these freeways. Where is the best location to put the airport so that they have to build the least amount of road? In the picture below, the blue lines are the proposed roads. After completing this Concept, you'll be able to use angle bisectors to help answer this question.
Watch This
CK12 Foundation: Chapter5AngleBisectorsA
James Sousa: Introduction to Angle Bisectors
James Sousa: Proof of the Angle Bisector Theorem
James Sousa: Proof of the Angle Bisector Theorem Converse
James Sousa: Solving For Unknown Values Using Angle Bisectors
Guidance
Recall that an angle bisector cuts an angle exactly in half. Let’s analyze this figure.
Angle Bisector Theorem: If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
In other words, if
Proof of the Angle Bisector Theorem:
Given:
Prove:
Statement  Reason 

1. 
Given 
2. 
Definition of an angle bisector 
3. 
Definition of perpendicular lines 
4. 
All right angles are congruent 
5. 
Reflexive PoC 
6. 
AAS 
7. 
CPCTC 
The converse of this theorem is also true.
Angle Bisector Theorem Converse: If a point is in the interior of an angle and equidistant from the sides, then it lies on the bisector of the angle.
Because the Angle Bisector Theorem and its converse are both true we have a biconditional statement. We can put the two conditional statements together using if and only if. A point is on the angle bisector of an angle if and only if it is equidistant from the sides of the triangle. Like perpendicular bisectors, the point of concurrency for angle bisectors has interesting properties.
Investigation: Constructing Angle Bisectors in Triangles
Tools Needed: compass, ruler, pencil, paper
1. Draw a scalene triangle. Construct the angle bisector of each angle. Use Investigation 14 and #1 from the Review Queue to help you.
Incenter: The point of concurrency for the angle bisectors of a triangle.
2. Erase the arc marks and the angle bisectors after the incenter. Draw or construct the perpendicular lines to each side, through the incenter.
3. Erase the arc marks from #2 and the perpendicular lines beyond the sides of the triangle. Place the pointer of the compass on the incenter. Open the compass to intersect one of the three perpendicular lines drawn in #2. Draw a circle.
Notice that the circle touches all three sides of the triangle. We say that this circle is inscribed in the triangle because it touches all three sides. The incenter is on all three angle bisectors, so the incenter is equidistant from all three sides of the triangle.
Concurrency of Angle Bisectors Theorem: The angle bisectors of a triangle intersect in a point that is equidistant from the three sides of the triangle.
If
In other words,
Example A
Is
In order for
Example B
If
Example C
\begin{align*}CB=BD\end{align*}
\begin{align*} x+7 &=2(3x4)\\ x+7 &=6x8\\ 15 &=5x\\ x &=3 \end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: Chapter5AngleBisectorsB
Concept Problem Revisited
The airport needs to be equidistant to the three highways between the three cities. Therefore, the roads are all perpendicular to each side and congruent. The airport should be located at the incenter of the triangle.
Vocabulary
An angle bisector cuts an angle exactly in half. Equidistant means the same distance from. A point is equidistant from two lines if it is the same distance from both lines. When we construct angle bisectors for the angles of a triangle, they meet in one point. This point is called the incenter of the triangle.
Guided Practice
1. Is there enough information to determine if \begin{align*}\overrightarrow{A B}\end{align*}
2. \begin{align*}\overrightarrow{MO}\end{align*}
3. A \begin{align*} 100^\circ\end{align*}
Answers:
1. No because \begin{align*}B\end{align*}
2. \begin{align*}LO = ON\end{align*}
\begin{align*}4x  5 &= 23\\ 4x &= 28\\ x &=7\end{align*}
3. We know that to bisect means to cut in half, so each of the resulting angles will be half of \begin{align*}100\end{align*}
Interactive Practice
Practice
For questions 16, \begin{align*}\overrightarrow{AB}\end{align*}
Is there enough information to determine if \begin{align*}\overrightarrow{AB}\end{align*}
 Fill in the blanks in the Angle Bisector Theorem Converse.
Given: \begin{align*}\overline{AD} \cong \overline{DC}\end{align*}
Prove: \begin{align*}\overrightarrow{BD}\end{align*}
Statement  Reason 

1.  
2.  The shortest distance from a point to a line is perpendicular. 
3. \begin{align*}\angle DAB\end{align*} 

4. \begin{align*}\angle DAB \cong \angle DCB\end{align*}  
5. \begin{align*}\overline{BD} \cong \overline{BD}\end{align*}  
6. \begin{align*}\triangle ABD \cong \triangle CBD\end{align*}  
7.  CPCTC 
8. \begin{align*}\overrightarrow{BD}\end{align*} bisects \begin{align*}\angle ABC\end{align*} 
Determine if the following descriptions refer to the incenter or circumcenter of the triangle.
 A lighthouse on a triangular island is equidistant to the three coastlines.
 A hospital is equidistant to three cities.
 A circular walking path passes through three historical landmarks.
 A circular walking path connects three other straight paths.
Multi Step Problem
 Draw \begin{align*}\angle ABC\end{align*} through \begin{align*}A(1, 3), B(3, 1)\end{align*} and \begin{align*}C(7, 1)\end{align*}.
 Use slopes to show that \begin{align*}\angle ABC\end{align*} is a right angle.
 Use the distance formula to find \begin{align*}AB\end{align*} and \begin{align*}BC\end{align*}.
 Construct a line perpendicular to \begin{align*}AB\end{align*} through \begin{align*}A\end{align*}.
 Construct a line perpendicular to \begin{align*}BC\end{align*} through \begin{align*}C\end{align*}.
 These lines intersect in the interior of \begin{align*}\angle ABC\end{align*}. Label this point \begin{align*}D\end{align*} and draw \begin{align*}\overrightarrow{BD}\end{align*}.
 Is \begin{align*}\overrightarrow{BD}\end{align*} the angle bisector of \begin{align*}\angle ABC\end{align*}? Justify your answer.