Explainer: Angle Bisector Theorem and Its Converse

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.

Let’s consider a triangle 𝐴𝐡𝐢 and the angle bisector of angle 𝐢, 𝐢𝐷.

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)

⃖⃗𝐴𝐸 is parallel to ⃖⃗𝐷𝐢. We have ∠𝐢𝐴𝐸=∠𝐷𝐢𝐴,∠𝐡𝐢𝐷=∠𝐢𝐸𝐴∠𝐡𝐢𝐷=∠𝐷𝐢𝐴𝐢𝐷𝐢.(alternateangles)(correspondingangles),and(bisectorofangle)

Therefore, ∠𝐢𝐴𝐸=∠𝐢𝐸𝐴, which means that 𝐴𝐢𝐸 is an isosceles triangle, and so 𝐢𝐸=𝐢𝐴.

As ⃖⃗𝐷𝐢 and ⃖⃗𝐴𝐸 are parallel, the triangles 𝐡𝐢𝐷 and 𝐡𝐸𝐴 are similar. It follows that 𝐡𝐷𝐡𝐴=𝐡𝐢𝐡𝐸, which can be rewritten as 𝐡𝐷𝐡𝐷+𝐷𝐴=𝐡𝐢𝐡𝐢+𝐢𝐸.

By multiplying both sides of the equation by (𝐡𝐢+𝐢𝐸)(𝐡𝐷+𝐷𝐴), we get 𝐡𝐷(𝐡𝐢+𝐢𝐸)=𝐡𝐢(𝐡𝐷+𝐷𝐴).

Expanding the brackets, we find 𝐡𝐷⋅𝐡𝐢+𝐡𝐷⋅𝐢𝐸=𝐡𝐢⋅𝐡𝐷+𝐡𝐢⋅𝐷𝐴.

Taking away 𝐡𝐷⋅𝐡𝐢, which is the same as 𝐡𝐢⋅𝐡𝐷, from both sides, we get 𝐡𝐷⋅𝐢𝐸=𝐡𝐢⋅𝐷𝐴.

Dividing now both sides by 𝐷𝐴⋅𝐢𝐸, we have 𝐡𝐷𝐷𝐴=𝐡𝐢𝐢𝐸.

And since 𝐢𝐸=𝐢𝐴, 𝐡𝐷=𝐷𝐡, and 𝐡𝐢=𝐢𝐡, we finally have 𝐷𝐡𝐷𝐴=𝐢𝐡𝐢𝐴.

Let’s see with an example how to apply this theorem.

Example 1: Applying the Angle Bisector Theorems

In the figure, 𝐴𝐷 bisects ∠𝐡𝐴𝐢, 𝐡𝐷=8, 𝐷𝐢=11, and the perimeter of △𝐴𝐡𝐢 is 57. Determine the lengths of 𝐴𝐡 and 𝐴𝐢.

  1. 𝐴𝐡=22, 𝐴𝐢=16
  2. 𝐴𝐡=16, 𝐴𝐢=22
  3. 𝐴𝐡=16, 𝐴𝐢=19
  4. 𝐴𝐡=19, 𝐴𝐢=22

Answer

𝐴𝐷 bisects ∠𝐴 in triangle 𝐴𝐡𝐢. From the angle bisector theorem (theorem I), we know that we have 𝐷𝐢𝐷𝐡=𝐴𝐢𝐴𝐡118=𝐴𝐢𝐴𝐡.

And we know as well that 𝐴𝐡+𝐡𝐷+𝐷𝐢+𝐴𝐢=57.

We know that 𝐡𝐷=8 and 𝐷𝐢=11, so 𝐴𝐡+8+11+𝐴𝐢=57𝐴𝐡=57βˆ’19βˆ’π΄πΆπ΄π΅=38βˆ’π΄πΆ.

By substituting this into our first equation, we get 𝐴𝐢38βˆ’π΄πΆ=1188⋅𝐴𝐢=11(38βˆ’π΄πΆ)8⋅𝐴𝐢=418βˆ’11⋅𝐴𝐢19⋅𝐴𝐢=418𝐴𝐢=41819𝐴𝐢=22.

And 𝐴𝐡=38βˆ’π΄πΆπ΄π΅=38βˆ’22=16.

Let’s look at another theorem, still in triangle 𝐴𝐡𝐢 whose angle bisector of angle 𝐢 is 𝐢𝐷. This theorem is about the length of 𝐢𝐷.

Angle Bisector Theorem (Part II)

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

Let’s see how to use this theorem.

Example 2: Applying the Angle Bisector Theorems

In the triangle 𝐴𝐡𝐢, 𝐴𝐡=76cm, 𝐴𝐢=57cm, and 𝐡𝐷=52cm. Given that 𝐴𝐷 bisects ∠𝐴 and intersects 𝐡𝐢 at 𝐷, determine the length of 𝐴𝐷.

Answer

𝐴𝐷 bisects ∠𝐴 in triangle 𝐴𝐡𝐢. From the angle bisector theorem (Part II), we know that we have 𝐴𝐷=βˆšπ΄πΆβ‹…π΄π΅βˆ’π·πΆβ‹…π·π΅.

From the question, we know the lengths of 𝐴𝐢, 𝐴𝐡, and 𝐷𝐡. Therefore, to find 𝐴𝐷, we must first find 𝐷𝐢. For this, we apply the angle bisector theorem (Part I): 𝐷𝐢𝐷𝐡=𝐴𝐢𝐴𝐡𝐷𝐢=𝐴𝐢𝐴𝐡⋅𝐷𝐡𝐷𝐢=5776β‹…52=39.cm

Now, we can plug in all the lengths: 𝐴𝐷=βˆšπ΄πΆβ‹…π΄π΅βˆ’π·πΆβ‹…π·π΅π΄π·=√57β‹…76βˆ’39β‹…52𝐴𝐷=48.cm

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 𝐷. 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:

Let’s see an example on how to use this theorem.

Example 3: Applying the Angle Bisector Theorems

Given that 𝐴𝐡=60, 𝐴𝐢=40, and 𝐡𝐢=31, what is 𝐢𝐷?

Answer

(𝐴𝐷) is the angle bisector of the exterior angle 𝐴 of 𝐴𝐡𝐢. Therefore, we have 𝐷𝐢𝐷𝐡=𝐴𝐢𝐴𝐡.

Since 𝐷𝐡=𝐷𝐢+𝐢𝐡, we can write 𝐷𝐢𝐷𝐢+𝐢𝐡=𝐴𝐢𝐴𝐡.

And by plugging in the lengths given in the question, 𝐷𝐢𝐷𝐢+31=4060.

By multiplying both sides by 60(𝐷𝐢+31), we get 60𝐷𝐢=40(𝐷𝐢+31)60𝐷𝐢=40𝐷𝐢+1,24020𝐷𝐢=1,240𝐷𝐢=1,24020=62.

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