Bisected Segments and Angles | Grade 8 Math | LevelKite

To bisect something means to cut it into two equal parts. In this lesson, you'll use that idea to find missing lengths and angle measures.

The midpoint of a segment is the point that bisects it, so the two parts on either side of it have the same length. Two segments with the same length are congruent. In a figure, matching tick marks on two segments state that they are congruent. The symbol $AB$ stands for the length of segment A B, and lengths in this lesson are in centimeters (cm).

The figures are not drawn to scale, so don't take any length or angle from how a drawing looks. Use only the marks and the facts stated in the text.

One tick on each part states that

AB = BC

, so B is the midpoint of segment A C. The marked length is segment A C, which measures 18 cm.

Worked example

Example 1: Halve a whole segment

In the figure, the ticks show that point B is the midpoint of segment A C. The marked length, segment A C, measures 18 cm. Find the length of segment A B.

The matching ticks state that segments A B and B C are congruent, so they have the same length. This fact comes from the marks, not from how long the parts look.
$AB = BC$
Point B lies on segment A C, so the two parts together make the whole. Write that sum, using the stated length of segment A C.
$AB + BC = 18$
Since BC equals AB, replace BC with AB. Replacing a quantity with an equal one keeps the equation true, and two copies of AB make 2 times AB.
$AB + AB = 2 \cdot AB = 18$
Divide both sides by 2 to find one part. Dividing both sides of an equation by the same nonzero number keeps the two sides equal.
$AB = \frac{18}{2} = 9$

Answer

Segment A B is $9$ cm long, and so is segment B C, because the ticks state that the two parts are equal.

You can bisect an angle too. An angle bisector is a ray that starts at the vertex of an angle and splits the angle into two angles of equal measure. Angles with equal measures are congruent, too. Angle measures are in degrees (°), and $m\angle ABD$ stands for the measure of angle A B D.

The figures in this lesson have no mark for equal angles, so a drawing can't show you a bisector. The text has to state it, as in ray B D bisects angle A B C. That sentence tells you that angle A B D and angle D B C have equal measures. Without it, you don't know the two angles are equal, however they look.

Ray B D is the bisector stated in the text, and it forms angle A B D and angle D B C. The marked angle is angle A B D, which measures

37^\circ

.

Worked example

Example 2: Double a part of an angle

In the figure, ray B D bisects angle A B C. The marked angle, angle A B D, measures $37^\circ$ . Find the measure of angle A B C.

The text states that ray B D bisects angle A B C, so the two angles it forms have equal measures.
$m\angle DBC = m\angle ABD = 37^\circ$
Angles A B D and D B C are adjacent and together make angle A B C, so their measures add. Replace each with 37 degrees; adding two equal parts doubles one part.
$m\angle ABC = m\angle ABD + m\angle DBC = 37^\circ + 37^\circ = 2 \cdot 37^\circ = 74^\circ$

Answer

Angle A B C measures $74^\circ$ , which is twice the $37^\circ$ part.

Key concept

Halve the whole, double a part

When tick marks show that a point is the midpoint of a segment, or when the text states that a ray bisects an angle, the whole is split into two congruent parts. Then each part is half of the whole, and the whole is twice one part. Every value comes from the text, which names each part by the figure's labels; nothing is measured from the drawing.

\text{part} = \frac{\text{whole}}{2} \qquad \text{whole} = 2 \cdot \text{part}

The ticks state that

BE = EA

, and the text states that ray B D bisects angle A B C. The marked values are segment B E, 6.5 cm, and angle A B C,

75^\circ

.

Worked example

Example 3: Decide whole or part first

In the figure, the ticks show that point E is the midpoint of segment B A, and ray B D bisects angle A B C. The marked length, segment B E, measures 6.5 cm. The marked angle, angle A B C, measures $75^\circ$ . Find the length of segment B A and the measure of angle D B C.

Sort the stated values first. Segment B E is one of the two ticked parts, so double it. Angle A B C is the whole angle, so halve it.
The ticks state that EA equals BE, and the two parts together make segment B A. Replace each part with 6.5.
$BA = BE + EA = 6.5 + 6.5 = 13$
Angles A B D and D B C add to angle A B C. The stated bisector makes them equal, so replace the measure of angle A B D with the measure of angle D B C.
$m\angle ABD + m\angle DBC = 75^\circ \;\Rightarrow\; 2 \cdot m\angle DBC = 75^\circ$
Divide both sides by the nonzero number 2. The result, 37.5, is exact, so no rounding is needed.
$m\angle DBC = \frac{75^\circ}{2} = 37.5^\circ$
Check by working backward: half of the whole segment should give the stated part, and twice the angle you found should give the stated whole.
$13 \div 2 = 6.5 \qquad 2 \cdot 37.5^\circ = 75^\circ$

Answer

Segment B A is $13$ cm long, and angle D B C measures exactly $37.5^\circ$ .

Takeaway: Bisecting makes two equal parts. Tick marks on the two parts of a segment, or a sentence stating that a ray bisects an angle, tell you that the parts are equal; how the drawing looks never does. Then decide whether the stated value is the whole or one part: halve the whole, and double a part.