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Triangle Angle Bisector Theorem

Triangle bisectors can be very helpful as we solve a wide range of geometrical problems. If we understand the triangle bisector theorem, we can use our knowledge of predetermined rules to quickly find missing values. But how exactly does the triangle bisector theorem work? What can it teach us about math? Let's find out:

The triangle bisector theorem, explained

The triangle angle bisector theorem states that:

• The angle bisector of a triangle's angle divides the opposite into two sections proportional to the other two sides

This might be a little tricky to understand by simply reading the explanation, so let's take a look at his handy diagram to help us visualize how the triangle angle bisector theorem works:

As we can see, our angle bisector goes from vertex A to point D. This bisector divides side CB into two different segments: CD and DB.

Now here's the crucial part:

If we compare BD to CD, we get a ratio that is identical to the ratio between AB and AC. The two relationships are proportional.

In other words, $\frac{\mathrm{BC}}{\mathrm{DC}}=\frac{\mathrm{AB}}{\mathrm{AC}}$

Proving that the triangle bisector theorem is correct

Remember, a theorem is something that we can prove. So let's see if we can check whether the triangle angle bisector theorem is valid.

Let's start by drawing a line that is parallel to AD. We can call this new line BE:

Let's also extend CA to meet our new parallel line at point E.

You may recall that the side-splitter theorem states that if one line is parallel to one side of a triangle and intersects two other sides, then it divides those sides proportionally.

In other words, $\frac{\mathrm{CD}}{\mathrm{DB}}=\frac{\mathrm{CA}}{\mathrm{AE}}$

This means that ∠4 and ∠1 are corresponding angles -- which means that they are congruent. Remember, corresponding angles are congruent in this situation.

AD is the angle bisector of the ∠CAB, so ∠2 is also congruent to ∠3.

Now we can apply the transitive property and state that ∠4 is congruent to ∠3. Remember the transitive property states that if two numbers are equal, and a third number is equal to one of the numbers, then the second number must also be equal to the third.

If ∠3 and ∠4 are congruent, then this must mean that ΔABE is an isosceles triangle. In other words, $\mathrm{AE}=\mathrm{AB}$ .

If we replace AE with AB in our earlier equation, we get:

$\frac{\mathrm{CD}}{\mathrm{DB}}=\frac{\mathrm{CA}}{\mathrm{AB}}$

Working with the triangle angle bisector theorem

Now that we know the triangle angle bisector theorem is valid, it's time to put our knowledge to good use.

Consider the following diagram:

Can we use the triangle angle bisector theorem to find the value of x?

We can see that there is an angle bisector that runs from vertex B to point D. This means that our theorem should work.

$\frac{\mathrm{AB}}{\mathrm{BC}}=\frac{\mathrm{AD}}{\mathrm{DC}}$

Now we can plug in our values:

$\frac{5}{12}=\frac{3.5}{x}$

If we cross-multiply, we get:

$5x=42$

Divide both sides by five, and we get:

$x=8.4$

Great! We have just used the triangle angle bisector theorem to determine that x is 8.4.

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