Angle Bisector
In short: An angle bisector is the ray leaving a vertex that carves a corner into two neighboring corners of identical size, each exactly half of the original.
GeometryGeometryG-CO.D.12
What is Angle Bisector?
An angle bisector works like a perfectly fair knife blade placed at the vertex of a corner. Each of the two pieces it makes has the same size, so each is half of the original.
Ray \(BD\) bisects \(\angle ABC\) when \(m\angle ABD = m\angle DBC = \frac{1}{2}\,m\angle ABC\). Any point sitting on this ray lies equally far from both sides of the corner, so three such rays in a triangle meet at the hub of the largest circle that fits inside it.
How to spot it
- A figure marks the two pieces of one corner with identical tick marks or identical little arcs.
- The text says bisects, splits in half or divides the corner into two equal parts.
- A triangle problem wants a point equally far from two sides, like a lamp that must light two walls evenly; think angle bisector.
- An expression is given for each half and you are told the halves match.
The method
- Set the two half-measures equal, because a bisector makes the pieces congruent.
- Solve the equation for the variable.
- Substitute the value back to find each half, then double it to get the whole corner.
- To build one with a compass: sweep an arc over both sides, then from each mark sweep matching arcs that cross, and rule a line from the vertex through that crossing.
Angle Bisector: solved example
Problem
Ray \(BD\) bisects \(\angle ABC\), with \(m\angle ABD = (4x + 7)^{\circ}\) and \(m\angle DBC = (6x – 13)^{\circ}\). Find \(x\) and \(m\angle ABC\).
Solution
The halves are equal: \(4x + 7 = 6x – 13\). Subtract \(4x\) and add 13 to get \(20 = 2x\), so \(x = 10\).
Each half is \(4(10) + 7 = 47\) and \(6(10) – 13 = 47\). The whole corner is \(47 + 47 = 94\), so \(m\angle ABC = \) \(94^{\circ}\).
Try it
Question 1
A \(128^{\circ}\) corner is bisected. What is each part?
Show the answer
\(128 \div 2 = 64\), so each part measures \(64^{\circ}\).
Question 2
Ray \(QS\) bisects \(\angle PQR\), which measures \(82^{\circ}\). If \(m\angle PQS = (3y + 8)^{\circ}\), find \(y\).
Show the answer
Half of 82 is 41, so \(3y + 8 = 41\), \(3y = 33\) and \(y = 11\).
Angle Bisector vs. Perpendicular Bisector
A bisector of a corner is a ray that halves a measure in degrees and starts at the vertex. A perpendicular bisector belongs to a segment: it cuts the length in half and also crosses it at \(90^{\circ}\). One divides an opening, the other divides a distance.
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