Angle Bisector Theorem - Proof, Converse, Formula, Examples

In geometry, the angle bisector of a triangle is a line or ray that divides the angle formed by two sides of the triangle into two equal angles. The angle bisector theorem states that the line bisecting an angle of a triangle also divides the opposite side into segments that are proportional to the other two sides.

To find the angle bisector of a triangle, you will need to know the lengths of all three sides of the triangle. You can then use the following steps:

1. Choose any two sides of the triangle and use the angle bisector theorem to find the length of the angle bisector for the angle between those two sides.

2. Draw the angle bisector for that angle.

3. Repeat the process for the other two angles of the triangle.

4. The three angle bisectors will intersect at a single point, which is called the incenter of the triangle.

The angle bisector theorem is often used to solve problems involving triangles, such as finding the length of a side or the measure of an angle. It can also be used to prove that certain triangles are similar.

The exterior angle bisector theorem

The exterior angle bisector theorem states that the angle bisector of an exterior angle of a triangle is also an angle bisector of the opposite interior angle. In other words, the angle bisector of an exterior angle of a triangle bisects the opposite interior angle into two equal angles.

To understand this theorem, it may be helpful to first review the definitions of the interior and exterior angles of a triangle. The interior angles of a triangle are the angles formed by two sides of the triangle and are always inside the triangle. The exterior angles of a triangle are the angles formed by one side of the triangle and an extension of another side and are always outside the triangle.

Here is an example of how the exterior angle bisector theorem works:

Imagine that you have a triangle with sides of lengths a, b, and c and interior angles A, B, and C. If you draw the angle bisector of the exterior angle C, it will also bisect the opposite interior angle A into two equal angles.

The exterior angle bisector theorem is often used in geometry to solve problems involving triangles, such as finding the length of a side or the measure of an angle. It can also be used to prove that certain triangles are similar.

The internal angle bisector theorem

The internal angle bisector theorem, also known as the triangle angle bisector theorem or the angle bisector theorem, states that the line bisecting an angle of a triangle also divides the opposite side into segments that are proportional to the other two sides. In other words, if you draw an angle bisector for an angle of a triangle, the line will also bisect the opposite side into segments that are proportional to the lengths of the other two sides.

To understand this theorem, it may be helpful to first review the definitions of the sides and angles of a triangle. In any triangle, there are three sides and three angles. The sides are labeled a, b, and c, and the angles are labeled A, B, and C.

Here is an example of how the internal angle bisector theorem works:

Imagine that you have a triangle with sides of lengths a, b, and c and interior angles A, B, and C. If you draw the angle bisector of angle A, it will also bisect the opposite side c into two segments, one of which is proportional to side a and the other of which is proportional to side b.

The internal angle bisector theorem is often used in geometry to solve problems involving triangles, such as finding the length of a side or the measure of an angle. It can also be used to prove that certain triangles are similar.

Angle bisector example

Here is an example of how to use the angle bisector theorem to find the length of a side of a triangle:

Imagine that you have a triangle with sides of lengths a = 6, b = 8, and c = 10 and interior angles A, B, and C. You are asked to find the length of side c.

To solve this problem, you can use the angle bisector theorem to find the ratio of the length of side c to the length of one of the other sides, a or b. For example, you could use the following steps:

1. Draw a diagram to represent the triangle.

2. Choose one of the angles of the triangle and draw the angle bisector for that angle.

3. Use the angle bisector theorem to find the ratio of the length of side c to the length of the other side. For example, if you choose angle B, you can use the following equation:

c/b = a/c

4. Solve the equation for the unknown side length. In this case, you can rearrange the equation to solve for c:

c = (a*b)/c

5. Substitute the known values for a and b into the equation and solve for c:

c = (6*8)/c

c = 48/c

c = 48/10

c = 4.8

6. Round the answer to the nearest whole number, since side lengths must be whole numbers:

c = 5

Therefore, the length of side c is 5.

This is just one example of how the angle bisector theorem can be used to find the length of a side of a triangle. You can use similar steps to solve other problems involving triangles.

Show that the angle bisectors of a triangle are concurrent

To show that the angle bisectors of a triangle are concurrent, you can use the angle bisector theorem and the fact that the sum of the interior angles of a triangle is 180 degrees.

Here are the steps to follow:

• Draw a diagram to represent the triangle.

• Choose any two angles of the triangle and draw the angle bisector for each angle.

• Use the angle bisector theorem to find the ratio of the lengths of the sides opposite the two angles to the lengths of the other two sides. For example, if you choose angles A and B, you can use the following equations:

c/b = a/c

a/c = b/a

• Solve the equations for the unknown side lengths. In this case, you can rearrange the equations to solve for a and b:

a = (c*b)/a

b = (a*c)/b

• Substitute the known values for c and the angle measures into the equations and solve for a and b.

• Use the fact that the sum of the interior angles of a triangle is 180 degrees to find the third angle measure.

• Show that the three angle measures add up to 180 degrees, which proves that the angle bisectors are concurrent.

For example, if the measures of angles A and B are 45 degrees and 60 degrees, respectively, you can use the following steps:

A = 45 degrees

B = 60 degrees

a = (c*b)/a

a= (c*60)/45

a = (4/3)c

b = (a*c)/b

b = (45*c)/60

b = (3/4)c

A + B + C = 180 degrees

45 + 60 + C = 180 degrees

C = 75 degrees

A + B + C = 180 degrees

45 + 60 + 75 = 180 degrees,

Therefore, the angle bisectors of the triangle are concurrent.

This is just one example of how to show that the angle bisectors of a triangle are concurrent. You can use similar steps to solve other problems involving triangles.

Prove that the angle bisectors of a triangle are concurrent

To prove that the angle bisectors of a triangle are concurrent, you can use the angle bisector theorem and the fact that the sum of the interior angles of a triangle is 180 degrees.

Here are the steps to follow:

• Draw a diagram to represent the triangle.

• Choose any two angles of the triangle and draw the angle bisector for each angle.

• Use the angle bisector theorem to find the ratio of the lengths of the sides opposite the two angles to the lengths of the other two sides. For example, if you choose angles A and B, you can use the following equations:

c/b = a/c

a/c = b/a

• Solve the equations for the unknown side lengths. In this case, you can rearrange the equations to solve for a and b:

a = (c*b)/a

b = (a*c)/b

• Substitute the known values for c and the angle measures into the equations and solve for a and b.

• Use the fact that the sum of the interior angles of a triangle is 180 degrees to find the third angle measure.

• Show that the three angle measures add up to 180 degrees, which proves that the angle bisectors are concurrent.

For example, if the measures of angles A and B are 45 degrees and 60 degrees, respectively, you can use the following steps:

A = 45 degrees

B = 60 degrees

a = (c*b)/a

a= (c*60)/45

a = (4/3)c

b = (a*c)/b

b = (45*c)/60

b = (3/4)c

A + B + C = 180 degrees

45 + 60 + C = 180 degrees

C = 75 degrees

A + B + C = 180 degrees

45 + 60 + 75 = 180 degrees,

Therefore, the angle bisectors of the triangle are concurrent.

This is just one example of how to show that the angle bisectors of a triangle are concurrent. You can use similar steps to solve other problems involving triangles.

External Bisector

The external bisector of a triangle is a line or ray that divides an exterior angle of a triangle into two equal angles. It is often used in geometry to solve problems involving triangles, such as finding the length of a side or the measure of an angle.

To find the external bisector of a triangle, you will need to know the lengths of all three sides of the triangle. You can then use the following steps:

• Choose an exterior angle of the triangle and draw the angle bisector for that angle.
• Use the angle bisector theorem to find the ratio of the length of the side opposite the exterior angle to the length of the other side.
• Solve the equation for the unknown side length.
• Substitute the known values for the other two side lengths into the equation and solve for the unknown side length. Round the answer to the nearest whole number, since side lengths must be whole numbers.

Here is an example of how to use the external bisector of a triangle to find the length of a side:

Imagine that you have a triangle with sides of lengths a = 6, b = 8, and c = 10 and exterior angle C. You are asked to find the length of side a.

To solve this problem, you can use the following steps:

• Draw a diagram to represent the triangle.
• Draw the external bisector for exterior angle C.
• Use the angle bisector theorem to find the ratio of the length of side a to the length of side b: a/b = c/a
• Solve the equation for the unknown side length: a = (c*b)/a
• Substitute the known values for b and c into the equation and solve for a: a = (10*8)/6 a = 16/2 a = 8 Therefore, the length of side is 8.

Angle bisector real life example

Here is an example of how the angle bisector theorem can be used in real life:

Imagine that you are a surveyor and you are asked to measure the height of a tall building. You are standing 100 meters away from the base of the building, and you have a surveyor's level, which allows you to measure the angle between the ground and the top of the building. The angle measure is 45 degrees.

To find the height of the building, you can use the angle bisector theorem as follows:

• Draw a diagram to represent the situation.
• The angle of elevation is formed by a horizontal line and a line of sight from a lower point to a higher point. Measure the distance between the two points.
• In this case, the distance is 100 meters.
• Measure the angle of elevation. In this case, the angle is 45 degrees.
• Use the tangent function to find the height of the building.
• You can use the following formula:
• height = horizontal distance * tan(angle)
• Substitute the known values into the equation and solve for the height: height = 100 * tan(45) height = 100 * 1 height = 100

Therefore, the height of the building is 100 meters.

This is just one example of how the angle bisector theorem can be used in real life. The theorem is often used in surveying and other fields to measure distances and angles.