Angle Bisector - Interactive Activity

Definitions and Main Ideas

An angle bisector splits an angle into two congruent angles, and, just like the perpendicular bisector, it carries a distance property: every point on the bisector is the same distance from the two sides of the angle. The three angle bisectors of a triangle meet inside the triangle at the incenter, which is equidistant from all three sides. Most problems in this lesson come down to one move — the two halves of a bisected angle have equal measures.

Vocabulary

Angle Bisector

A segment or ray that divides an angle into two congruent angles.

Equidistant

The same distance from two or more points or objects; a point on an angle bisector is equidistant from the sides of the angle.

Concurrent

Three or more lines or segments that intersect at a single point.

Incenter

The point where the angle bisectors of the three angles of a triangle meet; it is equidistant from the three sides of the triangle and always lies inside the triangle.

Main Ideas

1. An angle bisector creates two angles of equal measure, each half the measure of the original angle.
2. Any point on the bisector of an angle is equidistant from the sides of the angle.
3. A triangle has up to three angle bisectors, and they always intersect inside the triangle at the incenter.
4. The incenter is equidistant from the sides of the triangle.
5. When an angle and its half are both given as expressions, the whole angle equals twice the half — not the half itself.

Bisector Explorer

What to do: Move the whole-angle slider to change ∠ABC, and watch each half change with it. Move the point slider to slide point P along the bisector BD, and watch its perpendicular distances to the two sides. The two halves always stay equal, and the two distances always stay equal.

Incenter Explorer

What to do: Drag any vertex of the triangle and watch the three angle bisectors follow. They always intersect at the incenter I, the center of the circle that just touches all three sides. Use the preset buttons for clean examples, then drag to change the kind of triangle and watch where I stays.

Angle Bisector Challenge

What to do: Work each problem: find a half from the whole, the whole from a half, solve for x, or use the distance property. A first miss gives a hint and another try; a second miss shows the correct choice and explains it.

Build the Bisector

What to do: Work each problem one step at a time. First choose the statement or equation the problem calls for, then substitute or set the expressions equal, then solve and answer what was asked. The diagram labels a measure only after the step that finds it has been answered.

Your Turn

What to do: Answer each question. A first miss gives a hint and another try; a second miss shows the full explanation. Your score counts first-try answers only.