An angle bisector is a ray, line, or segment that divides an angle into two congruent angles. In a diagram of ∠ABC, if ray BD is the angle bisector, then ∠ABD = ∠DBC. This idea matters because it turns one angle measurement into two equal parts, making many geometry problems easier to solve.
Angle bisectors also connect angle relationships to side lengths in triangles.
Key Facts
- If BD bisects ∠ABC, then m∠ABD = m∠DBC.
- If m∠ABC = x and BD is the angle bisector, then m∠ABD = x/2 and m∠DBC = x/2.
- Angle Bisector Theorem: If AD bisects ∠A in triangle ABC and meets BC at D, then BD/DC = AB/AC.
- Converse Angle Bisector Theorem: If D is on BC and BD/DC = AB/AC, then AD bisects ∠A.
- Every point on an angle bisector is the same perpendicular distance from the two sides of the angle.
- A compass and straightedge construction of an angle bisector uses equal arcs from the angle sides, then draws a ray from the vertex through the arc intersection.
Vocabulary
- Angle bisector
- A ray, line, or segment that divides an angle into two congruent angles.
- Vertex
- The common endpoint of the two rays that form an angle.
- Congruent angles
- Angles that have exactly the same measure.
- Angle Bisector Theorem
- A triangle theorem stating that an angle bisector divides the opposite side into segments proportional to the adjacent sides.
- Perpendicular distance
- The shortest distance from a point to a line, measured along a segment that meets the line at a right angle.
Common Mistakes to Avoid
- Assuming a bisector splits the opposite side into equal lengths. This is only true in special cases, while the Angle Bisector Theorem gives a proportional relationship.
- Writing ∠ABD = ∠ABC when BD bisects ∠ABC. The bisector creates two smaller equal angles, so the correct statement is ∠ABD = ∠DBC.
- Forgetting that the vertex must stay the same when naming the original angle. In ∠ABC, B is the vertex, so the bisector must start at B.
- Using a ruler alone to construct an angle bisector. A correct compass and straightedge construction depends on equal arcs, not guessed midpoint measurements.
Practice Questions
- 1 Ray BD bisects ∠ABC. If m∠ABC = 74°, find m∠ABD and m∠DBC.
- 2 In triangle ABC, AD bisects ∠A and meets BC at D. If AB = 12, AC = 18, and BD = 8, find DC using BD/DC = AB/AC.
- 3 A point P lies inside an angle and is the same perpendicular distance from both sides of the angle. Explain what this tells you about the location of P.