In this explainer, we will learn how to use the theorem of angle bisector and its converse to find a missing side length in a triangle.

Letβs consider a triangle and the angle bisector of angle , .

### Angle Bisector Theorem (Part I)

If an interior angle of a triangle is bisected, the bisector divides the opposite side into segments whose lengths are in the same ratio as the lengths of the other sides of the triangle.

That is,

### How to Prove the Angle Bisector Theorem (Part I)

is parallel to . We have

Therefore, , which means that is an isosceles triangle, and so .

As and are parallel, the triangles and are similar. It follows that which can be rewritten as

By multiplying both sides of the equation by , we get

Expanding the brackets, we find

Taking away , which is the same as , from both sides, we get

Dividing now both sides by , we have

And since , , and , we finally have

Letβs see with an example how to apply this theorem.

### Example 1: Applying the Angle Bisector Theorems

In the figure, bisects , , , and the perimeter of is 57. Determine the lengths of and .

- ,
- ,
- ,
- ,

### Answer

bisects in triangle . From the angle bisector theorem (theorem I), we know that we have

And we know as well that

We know that and , so

By substituting this into our first equation, we get

And

Letβs look at another theorem, still in triangle whose angle bisector of angle is . This theorem is about the length of .

### Angle Bisector Theorem (Part II)

In any triangle , if is the angle bisector of angle , then we have

Letβs see how to use this theorem.

### Example 2: Applying the Angle Bisector Theorems

In the triangle , , , and . Given that bisects and intersects at , determine the length of .

### Answer

bisects in triangle . From the angle bisector theorem (Part II), we know that we have

From the question, we know the lengths of , , and . Therefore, to find , we must first find . For this, we apply the angle bisector theorem (Part I):

Now, we can plug in all the lengths:

There is a third angle bisector theorem, which concerns the bisector of an exterior angle of a triangle.

### Angle Bisector Theorem (Part III)

The bisector of an exterior angle of a triangle intersects an extension of the opposite side of the triangle at . The ratio of the length of only the extension of the opposite side to the length of the opposite side plus the extension is equal to the ratio of the lengths of the other two sides of the triangle:

Letβs see an example on how to use this theorem.

### Example 3: Applying the Angle Bisector Theorems

Given that , , and , what is ?

### Answer

is the angle bisector of the exterior angle of . Therefore, we have

Since , we can write

And by plugging in the lengths given in the question,

By multiplying both sides by , we get