Angle Bisector Theorem
Angle bisector theorem states that an angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. An angle bisector is a ray that divides a given angle into two angles of equal measures. Let us learn more about the angle bisector theorem in this article.
| 1. | What is Angle Bisector Theorem? |
| 2. | Angle Bisector Theorem Proof |
| 3. | Converse of Angle Bisector Theorem |
| 4. | Angle Bisector Theorem Formula |
| 5. | FAQs on Angle Bisector Theorem |
What is Angle Bisector Theorem?
The triangle angle bisector theorem states that in a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle. Consider the figure below:
Here, PS is the bisector of ∠P. According to the angle bisector theorem, PQ/PR = QS/RS or a/b = x/y.
An angle bisector is a line or ray that divides an angle in a triangle into two equal measures. The main properties of an angle bisector are that any point on the bisector of an angle is equidistant from the sides of the angle and the angle bisector divides the opposite side of a triangle in the ratio of the adjacent sides, which is known as the angle bisector property of triangle.
Angle Bisector Theorem Proof
Statement: In a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle. Let us see the proof of this.
Draw a ray CX parallel to AD, and extend BA to intersect this ray at E.
By the basic proportionality theorem, we have that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
In ΔCBE, DA is parallel to CE.
BD/DC = BA/AE ⋯ (1)
Now, we are left with proving that AE = AC.
Let's mark the angles in the above figure.
Since DA is parallel to CE, we have
∠DAB = ∠CEA (corresponding angles) ----- (2)
∠DAC = ∠ACE (alternate interior angles) ----- (3)
Since AD is the bisector of ∠BAC, we have ∠DAB = ∠DAC ---- (4).
From (2), (3), and (4), we can say that ∠CEA = ∠ACE. It makes ΔACE an isosceles triangle. Since sides opposite to equal angles are equal, we have AC = AE.
Substitute AC for AE in equation (1).
BD/DC = BA/AC
Hence proved.
Converse of Angle Bisector Theorem
The converse of angle bisector theorem states that if the sides of a triangle satisfy the following condition "If a line drawn from a vertex of a triangle divides the opposite side into two parts such that they are proportional to the other two sides of the triangle", it implies that the point on the opposite side of that angle lies on its angle bisector. Here, it is known to us that sides are in proportion, and from this, we came to a conclusion that the line/ray/segment is the angle bisector of the respective angle. This is known as the converse of angle bisector theorem in geometry.
Look at the image below to understand it visually.
Angle Bisector Theorem Formula
Triangle angle bisector theorem states that "In a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle". From this, we can write the triangle angle bisector theorem formula as,
\(\dfrac{\text{BD}}{\text{DC}} = \dfrac{\text{AB}}{\text{AC}}\)
► Related Topics
Listed below are a few interesting topics related to the triangle angle bisector theorem, take a look.
Angle Bisector Theorem Examples
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Example 3: Look at ΔABC shown below.
If BD bisects ∠B, can you find the value of x?
Solution:
Given that, BD is the bisector of ∠B.
According to the triangle angle bisector theorem,
AB/BC = AD/DC
x/(x-2) = (x+2)/(x-1)
x(x−1) = (x−2)(x+2)
x2 − x = x2 − 4
−x = −4
x = 4
Therefore, the value of x is 4.
FAQs on Angle Bisector Theorem
What is the Triangle Angle Bisector Theorem?
The triangle angle bisector theorem states that "The bisector of any angle inside a triangle divides the opposite side into two parts proportional to the other two sides of the triangle which contain the angle."
What is the Formula for Angle Bisector Theorem?
Let AD be the bisector of ∠A in ΔABC. According to the angle bisector theorem formula, BD/DC = AB/AC.
How are the Side-Splitter Theorem and the Angle Bisector Theorem Similar?
The only similarity between the side-splitter theorem and the angle bisector theorem is that both the theorems are related to the proportions of side lengths of the triangle.
How to Use Angle Bisector Theorem?
The triangle angle bisector theorem can be used to find the missing lengths of the sides of a triangle. It establishes a relation between the sides.
What is the Converse of the Angle Bisector Theorem?
If a line or a ray AD is drawn in ΔABC such that BD/DC = AB/AC, then AD bisects the ∠A. This is the statement of angle bisector theorem converse.
How to Prove Angle Bisector Theorem?
To prove the angle bisector theorem, we need to extend the sides of the triangle and make another triangle right next to it. Then, we use the basic proportionality theorem to state the relationship between the sides of the triangle drawn.