Key takeaway: An angle bisector is a ray from the vertex that cuts an angle into two angles of equal measure, so each piece is exactly half of the original opening.
Grade 10 – GeometryG-CO.D.12Geometry and Measurement
Definition of Angle Bisector
Fold a paper corner so that its two edges lie on top of each other; the crease is an angle bisector. It starts at the vertex and runs through the inside of the opening, leaving two smaller angles that match in size.
Every point on the crease is the same distance from both edges. That is why the three creases of a triangle meet at the incenter, the center of the largest circle that fits inside. A compass and straightedge can draw the crease without any measuring.
Formula and notation
If ray \(\vec{QS}\) bisects \(\angle PQR\), then:
\[ m\angle PQS = m\angle SQR = \tfrac{1}{2}\, m\angle PQR \]
Matching arcs or tick marks in a diagram show that the two small angles are congruent.
Key facts
- The two pieces are congruent, and each is half the size of the whole opening.
- Points on the crease are equidistant from the two sides of the opening.
- A triangle has three of them, and they all pass through a single point.
- Halving a 180 degree opening gives two right angles.
Angle Bisector example
A kite frame has a corner at vertex \(B\). Ray \(\vec{BD}\) is the angle bisector of \(\angle ABC\). The pieces measure \(m\angle ABD = (7y + 3)^{\circ}\) and \(m\angle DBC = (9y – 11)^{\circ}\). How wide is the whole corner?
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The pieces must match: \(7y + 3 = 9y – 11\).
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Move terms: \(14 = 2y\), so \(y = 7\).
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One piece: \(7(7) + 3 = 52\). The other: \(9(7) – 11 = 52\). Add them.
The corner \(\angle ABC\) is \(104^{\circ}\) wide.
Common mistake
Watch out
A student sees a ray inside a \(60^{\circ}\) corner that makes a \(25^{\circ}\) piece and a \(35^{\circ}\) piece, and calls it the bisector because it lies between the sides.
Why it is wrong
Lying inside is not enough. A bisector must create two equal pieces, here \(30^{\circ}\) and \(30^{\circ}\). Since \(25 \neq 35\), that ray is not a bisector.
Related terms and lessons
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