Bisector of a Given Angle

An angle bisector is a ray that divides an angle into two congruent angles. In other words, it’s a line (or a segment of a line) that cuts an angle exactly in half. This means the two angles formed by the bisector have the same measure.

The angle bisector starts at the vertex (the point where the two sides of the angle meet) and extends infinitely in one direction. When the angle is bisected, it creates two smaller angles, each with an angle measurement that’s half the size of the original angle.

Formulae

There isn’t a single, specific formula for calculating an angle bisector itself. However, understanding angle bisectors relies on formulas related to angles and geometry:

• If we know the measure of the original angle, we can find the measure of each smaller angle formed by the bisector: $ \text{Angle of each smaller angle} = \frac{\text{Original Angle}}{2} $
• In a triangle, the Angle Bisector Theorem helps relate the lengths of sides to the segments created by the angle bisector on the opposite side. If AD is the angle bisector of $ \angle BAC $ in $ \triangle ABC $, and D is on BC, then: $ \frac{AB}{AC} = \frac{BD}{DC} $
• Trigonometric formulas are sometimes required for angle bisectors, primarily when calculating the lengths of the segments created in triangle. For example the length of the bisector d in $ \triangle ABC $ from vertex A can be calculated by: $ d^2 = ab – \frac{abcd}{(a+b)^2} $ where a and b are the lengths of AB and AC respectively, and c and d are the length of the segments of BC divided by the angle bisector.

Examples

Example-1: If an angle measures $80^\circ$, its angle bisector divides it into two angles, each measuring:

$ \frac{80^\circ}{2} = 40^\circ $

Example-2: In $ \triangle ABC $, $ \angle B $ is bisected by BD. If $ AB = 6 $, $ BC = 8 $, and $ AC $ is divided into two segments with lengths $ x $ and $ 12-x $ , using the Angle Bisector Theorem: $ \frac{6}{8} = \frac{x}{12-x} $. Solving for $ x $, we get $ x = \frac{36}{7} $. Therefore the sides are divided into $ \frac{36}{7} $ and $ 12 – \frac{36}{7} = \frac{48}{7} $

Theorem with Proof

Theorem: Any point on the angle bisector is equidistant from the sides of the angle.

Proof:

1. Let $ \angle BAC $ be the angle, and AD be its bisector. Let P be any point on AD.
2. Draw perpendiculars from P to AB and AC, and label the points of intersection as E and F respectively. (PE is perpendicular to AB, and PF is perpendicular to AC)
3. We now have two right triangles: $ \triangle PAE $ and $ \triangle PAF $.
4. $ \angle PAE = \angle PAF $ because AD is the angle bisector.
5. $ PA $ is common to both triangles.
6. $ \angle PEA = \angle PFA = 90^\circ $ (by construction).
7. By the Angle-Angle-Side (AAS) congruence criterion, $ \triangle PAE \cong \triangle PAF $.
8. Therefore, $ PE = PF $ (corresponding parts of congruent triangles are equal).
9. This means the distance from point P to AB (PE) is equal to the distance from point P to AC (PF). Since P was any point on the bisector, this holds true for all points.

Common mistakes by students

• Confusing the angle bisector with the perpendicular bisector: Students often mistakenly believe that the angle bisector and the perpendicular bisector are the same thing. The angle bisector divides an angle, while the perpendicular bisector divides a line segment at a right angle.
• Incorrectly applying the Angle Bisector Theorem: Students might misapply the ratios in the Angle Bisector Theorem or use incorrect side lengths.
• Failing to understand the concept of equidistance: Students may not grasp that every point on the angle bisector is equidistant from the sides of the angle.
• Incorrectly calculating angles: Miscalculating the angle measurements of the bisected angles or neglecting to half the original angle.

Real Life Application

Angle bisectors have various applications in real life:

• Architecture and Design: Used to create symmetrical designs and to ensure that angles are accurately divided.
• Navigation and Surveying: Used to divide angles in survey maps and in triangulation to calculate distances.
• Engineering: Used in designing symmetrical objects and in the construction of bridges and other structures.
• Robotics: Used in path planning to make a robot’s movement symmetric.

Fun Fact

The concept of angle bisectors is closely related to the concept of symmetry. The angle bisector acts as an axis of symmetry for the angle itself. If you were to fold the angle along its bisector, the two sides would perfectly overlap.

Recommended YouTube Videos for Deeper Understanding

Q.1 The angle bisector of $\angle ABC$ divides the angle into two angles of what measure if $\angle ABC = 60^\circ$?

Check Solution

Ans: B

The angle bisector divides the angle into two equal parts. So, each angle will be $\frac{60^\circ}{2} = 30^\circ$.

Q.2 In triangle $PQR$, the angle bisector of $\angle P$ intersects $QR$ at point $S$. If $\angle P = 100^\circ$, what is the measure of $\angle RPS$?

Check Solution

Ans: B

The angle bisector divides $\angle P$ into two equal angles. Therefore, $\angle RPS = \frac{100^\circ}{2} = 50^\circ$.

Q.3 The angle bisectors of two adjacent angles on a straight line always form an angle of:

Check Solution

Ans: C

Adjacent angles on a straight line sum to $180^\circ$. Their bisectors will form an angle of $\frac{180^\circ}{2} = 90^\circ$.

Q.4 If the angle bisector of $\angle XYZ$ is $YV$, and $\angle XYV = 25^\circ$, what is the measure of $\angle XYZ$?

Check Solution

Ans: B

Since $YV$ bisects $\angle XYZ$, then $\angle XYZ = 2 \times \angle XYV = 2 \times 25^\circ = 50^\circ$.

Q.5 In a triangle $ABC$, if the angle bisectors of $\angle B$ and $\angle C$ intersect at point $I$, and $\angle BIC = 120^\circ$, what is the measure of $\angle A$?

Check Solution

Ans: D

We know that $\angle BIC = 180^\circ – \frac{\angle B}{2} – \frac{\angle C}{2} = 120^\circ$. Thus, $\frac{\angle B}{2} + \frac{\angle C}{2} = 60^\circ$, and $\angle B + \angle C = 120^\circ$. Since $\angle A + \angle B + \angle C = 180^\circ$, then $\angle A = 180^\circ – (\angle B + \angle C) = 180^\circ – 120^\circ = 60^\circ$.

Next Topic: Perpendicular Bisector of a Given Line Segment