Using Angle Bisectors of Triangles

Using Angle Bisectors of Triangles

Geometry

Mrs. LaGrange

Recall:

As we move through this lesson keep in mind the things that you learned about perpendicular bisectors. You will notice similarities and differences.

Here's a quick at a glance to help you out.

Math Spoken Here!

Angle Bisector: an angle bisector is a ray that divides an angle into two congruent adjacent angles.  $\overline{BD}$ is the angle bisector.

Distance from a point to a line: The length of the perpendicular segment from the point to the line. (red dashed line)

Need to know...

Angle Bisector Theorem:

If a point is on the bisector of an angle, it is equidistant from the sides of the angle.

Note that this theorem allows us to find side length of the perpendicular lines if we know the angles are congruent.

Need to know...

Converse of the Angle Bisector Theorem:

If a point is on the interior of an angle and is equidistant from the two sides of an angle, it is on the angle bisector.

Note that this theorem allows us to find angle measures of the bisected angle if we know the perpendicular lines are congruent.

Example:

Solution: 45.

Explanation:  Since  $\overline{CB}\perp\overline{CD}$  and  $\overline{BE}\perp\overline{DE}$  and  $CB=BE=7$ ,  $\overrightarrow{DB}$  bisects  $\angle CDE$  by the Converse of the Angle Bisector Theorem. So,  $m\angle BDC=m\angle BDE\ =45\degree.$  

Example:

EF = 87 by the Angle Bisector Theorem.

Multiple Choice

The figure is an example of a(n) ...

• angle bisector
• perpendicular bisector
• median
• midsegment

Multiple Choice

What type of angles are these?

• Adjacent Angles

• Linear Pair

• Complementary Angles

• Supplementary Angles

• Vertical Angles

Multiple Choice

Find m< JKM

• 90
• 76
• 38
• 83

Example: Using Algebra

Ray CE bisects angle BCD by the converse of the angle bisector theorem, angle BCE is congruent to angle DCE, so:

a+26=2a

26 = a

Example: Using Algebra Side Length

By the angle bisector theorem, we know that GH=HE.

u+26=2u

26=u

Multiple Choice

Find the measure of angle BAC.

Multiple Choice

Find LK

• 3
• 5
• -3
• 6

Math Spoken Here!

The point of concurrency of angle bisectors of a triangle is called the incenter.

P is equidistant from each side of the triangle.

Math Spoken Here!

The incenter always lies inside the triangle.

The incenter is equidistant from all three sides of the triangle.

It is the center of a circle inscribed within the triangle that touches all three sides.

Multiple Choice

Name the point of concurrency shown.

• Circumcenter
• Incenter
• Supercenter
• Neither

Multiple Choice

Which point is equidistant from the sides?

• Circumcenter
• Incenter
• Neither
• Bilateral

Multiple Choice

Which of the images represents the Incenter of a Triangle

• First
• Second
• Third

Multiple Choice

Define: INCENTER OF A TRIANGLE

• the intersection point of the three angle bisectors of a triangle 
• when three or more lines intersect at a single point 
• the intersection point of the three perpendicular bisectors of a triangle 
• the point of intersection of three or more lines 

Great job!

Your assignment for today is IXL Geometry M2. Strive for 85!

M2 has a combination of questions from perpendicular bisectors to angle bisectors. Feel free to look back at the previous lesson if you have forgotten something.