Angle Bisector

Angle Bisector

A line or ray that divides an angle in half. For polygons, an angle bisector is a line that bisects an interior angle.

Note: The angle bisectors of a triangle are concurrent and intersect at the incenter.

Angle bisector of ∠ABC
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See also

Key Formula

\text{If ray } BD \text{ bisects } \angle ABC, \text{ then } \angle ABD = \angle DBC = \frac{1}{2}\,\angle ABC

Where:

$\angle ABC$ = The original angle being bisected
$BD$ = The bisecting ray drawn from vertex B
$\angle ABD$ = One of the two equal angles formed
$\angle DBC$ = The other equal angle formed

Worked Example

Problem: Ray BD bisects angle ABC, which measures 74°. If angle ABD is expressed as (3x + 5)°, find the value of x.

Step 1: Since BD bisects angle ABC, each half equals half of 74°.

\angle ABD = \angle DBC = \frac{74°}{2} = 37°

Step 2: Set the expression for angle ABD equal to 37° and solve for x.

3x + 5 = 37

Step 3: Subtract 5 from both sides.

3x = 32

Step 4: Divide both sides by 3.

x = \frac{32}{3} \approx 10.\overline{6}

Answer: x = 32/3, which is approximately 10.67. Each half of the bisected angle measures 37°.

Another Example

Problem: In triangle PQR, the angle bisector of angle P meets side QR at point S. If PQ = 12, PR = 8, and QR = 15, find the lengths QS and SR using the Angle Bisector Theorem.

Step 1: The Angle Bisector Theorem states that the bisector from P divides the opposite side QR in the ratio of the adjacent sides.

\frac{QS}{SR} = \frac{PQ}{PR} = \frac{12}{8} = \frac{3}{2}

Step 2: Let QS = 3k and SR = 2k. Since QS + SR = QR = 15, solve for k.

3k + 2k = 15 \implies 5k = 15 \implies k = 3

Step 3: Calculate each segment length.

QS = 3(3) = 9, \quad SR = 2(3) = 6

Answer: QS = 9 and SR = 6. The angle bisector from P splits the opposite side in a 3 : 2 ratio, matching the ratio of the two adjacent sides.

Frequently Asked Questions

How do you construct an angle bisector with a compass and straightedge?

Place the compass point at the vertex and draw an arc that crosses both sides of the angle. From each intersection point, draw two arcs of equal radius so they cross each other inside the angle. Draw a ray from the vertex through the point where these two arcs meet. This ray is the angle bisector.

What is the Angle Bisector Theorem?

The Angle Bisector Theorem states that in a triangle, the bisector of any angle divides the opposite side into two segments whose lengths are proportional to the two adjacent sides. If the bisector of angle A in triangle ABC meets side BC at point D, then BD/DC = AB/AC.

Angle Bisector vs. Perpendicular Bisector

An angle bisector splits an angle into two equal halves and originates from a vertex. A perpendicular bisector splits a line segment into two equal halves at a 90° angle. In a triangle, angle bisectors meet at the incenter (center of the inscribed circle), while perpendicular bisectors of the sides meet at the circumcenter (center of the circumscribed circle). They serve different geometric purposes despite both involving the idea of bisecting.

Why It Matters

Angle bisectors are essential in triangle geometry because the three angle bisectors of any triangle always meet at a single point called the incenter, which is the center of the triangle's inscribed circle. This property is used in engineering, architecture, and design whenever you need to find the largest circle that fits inside a triangular region. The Angle Bisector Theorem also provides a powerful tool for finding unknown side lengths in triangles without needing trigonometry.

Common Mistakes

Mistake: Confusing the angle bisector with the median or the perpendicular bisector of the opposite side.

Correction: A median connects a vertex to the midpoint of the opposite side, and a perpendicular bisector is perpendicular to a side at its midpoint. An angle bisector divides the angle at the vertex into two equal parts. These three lines generally go in different directions and meet the opposite side at different points.

Mistake: Assuming the angle bisector always hits the midpoint of the opposite side.

Correction: The angle bisector hits the midpoint of the opposite side only when the triangle is isosceles (with the two adjacent sides equal). In general, the Angle Bisector Theorem tells you the bisector divides the opposite side in the ratio of the adjacent sides, not at the midpoint.

Related Terms

Angle — The geometric object the bisector divides
Bisect — General term meaning to cut in half
Incenter — Point where triangle's angle bisectors meet
Inscribed Circle — Circle centered at the incenter, tangent to all sides
Concurrent — Property of angle bisectors meeting at one point
Ray — The geometric form an angle bisector takes
Triangle — Common shape where angle bisectors are studied
Interior Angle — Angle inside a polygon that can be bisected