Perpendicular and angle bisectors: How to find lengths and angles

A bisector splits something into two equal parts. When it bisects an angle, the two angle measures match; when it perpendicularly bisects a segment, the two lengths match and the lines meet at a right angle. Let’s use both ideas in a few examples.

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An angle bisector makes two equal angles

An angle bisector starts at the vertex and divides the angle into two congruent angles. If the ray from $A$ through $D$ bisects $\angle CAB$, then both smaller angles measure half of $\angle CAB$.

$\begin{array}{l}m\angle CAD=m\angle DAB\\ =\frac{1}{2}m\angle CAB\end{array}$

Example: Find the third angle of a triangle

In the figure, $AD$ bisects the angles at $A$ and $D$. We’re given $m\angle DAB={31}^{°}$ and $m\angle BDC={66}^{°}$. Find $m\angle DCA$.

The two angles in triangle ACD that meet segment AD are 31° and 33°.

Since $AD$ bisects the angle at $A$, $m\angle CAD={31}^{°}$. At $D$, the bisector cuts ${66}^{°}$ in half, so $m\angle ADC={33}^{°}$. Those are two of the angles in triangle $ACD$.

$\begin{array}{l}{31}^{°}+{33}^{°}+m\angle DCA={180}^{°}\\ m\angle DCA={116}^{°}\end{array}$

A perpendicular bisector meets a segment at its midpoint

The midpoint gives us two equal lengths. Perpendicular tells us that the angle where the lines cross is 90°. Use whichever fact gives you an equation for the quantity you’re finding.

Example: Set the two lengths equal

Segment $CM$ meets $AJ$ at a right angle at its midpoint $M$. We know that $AM=5y+8$ and $MJ=8.2y$.

The matching tick marks show that AM and MJ have equal lengths.

Because $M$ is the midpoint, $AM=MJ$. Now we can solve for $y$.

$\begin{array}{l}5y+8=8.2y\\ 8=3.2y\\ y=2.5\end{array}$

With $y=2.5$, both expressions give the same length, $20.5$.

Example: Use the right angle at the midpoint

Segment $XM$ is perpendicular to $ZY$ at its midpoint. The angle $\angle XYM$ measures ${36}^{°}$. To find $m\angle YXM$, use the angle sum in triangle $XMY$.

Use the 90° angle at M and the 36° angle at Y to find the angle at X.

$\begin{array}{l}{90}^{°}+{36}^{°}+m\angle YXM={180}^{°}\\ m\angle YXM={54}^{°}\end{array}$

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