Angle Bisector Theorem

What is the Angle Bisector Theorem?

The angle bisector theorem states that if an angle is bisected, then the bisector is the perpendicular bisector of the angle.

Angle Bisector Theorem Statement:

The angle bisector theorem states that the angle bisector of an angle is the line that divides the angle into two equal parts.

Proof:

Given:

Angle ABC is bisected by line segment AB

Angle ACB is equal to angle ABC

Since angles ACB and ABC are equal, line segment AB is also the angle bisector of angle ACB.

Explanation of what is Angle Bisector Theorem Converse:

The angle bisector theorem states that if two lines intersect at an angle and one of the lines is perpendicular to the other, then the angle bisector of the angle is perpendicular to the other line. The converse of the angle bisector theorem states that if two lines intersect at an angle and one of the lines is perpendicular to the other, then the angle bisector of the angle is also perpendicular to the other line.

Converse of Angle Bisector Theorem Statement:

If two angles are bisected by a line, then the line is the angle bisector.

Converse: If a line is the angle bisector, then the two angles are bisected by the line.

What is the History of the Angle Bisector Theorem?

The angle bisector theorem is a geometric theorem that states that if an angle is bisected, then the line segment connecting the two points of intersection is the angle’s bisector. This theorem is often used in proofs involving triangles.

What is the Application of the Angle Bisector Theorem?

The angle bisector theorem states that if an angle is bisected, then the two angles formed are congruent.

How is the Converse of the Angle Bisector Theorem Applied If the Point is in the Exterior of the Angle?

If the point is in the exterior of the angle, then the converse of the angle bisector theorem is not applied.

How to State and Prove Angle Bisector Theorem:

Given a triangle, and two points on the triangle that are not on the same line as the triangle’s base, there is a line segment that connects those two points and is perpendicular to the triangle’s base. This line segment is the angle bisector of the triangle.

Given Data to State and Prove Angle Bisector Theorem:

Given: A line segment AB with endpoints A(0,0) and B(3,4)

The angle bisector of the angle AOB is the line segment CD with endpoints C(1,2) and D(2,5).

Proof:

Given: A line segment AB with endpoints A(0,0) and B(3,4)

The angle bisector of the angle AOB is the line segment CD with endpoints C(1,2) and D(2,5).

We will use the following theorem to prove that CD is the angle bisector.

Theorem:

If two points lie on a line, then the line segment connecting them is the angle bisector of the angle formed by the points.

Proof:

Given: A line segment AB with endpoints A(0,0) and B(3,4)
The angle bisector of the angle AOB is the line segment CD with endpoints C(1,2) and D(2,5).

We will use some basic algebra to show that CD is the angle bisector.

Angle AOB = Angle AOC + Angle COB

Since Angle AOC and Angle COB are both right angles, we can use the Pythagorean theorem to show that:

cos(AOC) = cos(COB)

Angle Bisector Theorem Proof:

Angle bisectors are straight lines that intersect at the midpoint of the angle and divide it into two equal angles.

The following theorem states that the angle bisector of an angle is perpendicular to the side of the angle opposite the angle.

Theorem:

The angle bisector of an angle is perpendicular to the side of the angle opposite the angle.

Proof:

Let ABC be an angle with bisector BD.

Since BD is the angle bisector, it divides ABC into two equal angles, DEF and DGF.

Angle DEF is opposite the side AB, and angle DGF is opposite the side BC.

Since both angles are equal, DF must be perpendicular to BC.

This proves that the angle bisector of an angle is perpendicular to the side of the angle opposite the angle.

Angle Bisector Theorem Examples:

1. In the figure, the angle bisector of ∠A is line segment AB.

2. In the figure, the angle bisector of ∠A is line segment BC.

3. In the figure, the angle bisector of ∠A is line segment CD.

4. In the figure, the angle bisector of ∠A is line segment AD.

:

– “He who increaseth knowledge, increaseth sorrow.” (Ecclesiastes 1:18)

– “For in much wisdom is much grief: and he that increaseth knowledge increaseth sorrow.” (Proverbs 1:18)