Bisectors: Splitting Segments and Angles in Half | Grade 7 Math | LevelKite

Fold a strip of paper so that its two ends meet. The crease splits the strip into two equal parts. To bisect something means exactly that: to split it into two equal parts.

In this lesson you will work with two things you can bisect: a segment, which is the straight path between two points, and an angle, whose size you already know as a measure in degrees. You will not measure anything here. Each problem states the lengths and angle measures you need, and your job is to use the meaning of bisect to find the ones that are missing.

The two points at the ends of a segment are its endpoints, and they give the segment its name: segment $AB$ runs from point $A$ to point $B$ . In this lesson, two letters written on their own, such as $AB$ , stand for the length of that segment.

The midpoint of a segment is the point on the segment that is the same distance from both endpoints. If your folded paper strip is $10$ cm long, the crease is $5$ cm from each end, so the crease marks the midpoint. The midpoint bisects the segment.

If $M$ is the midpoint of segment $AB$ , two facts are true:

The two parts are equal: $AM = MB$ .
The two parts make the whole: $AM + MB = AB$ .

A ray is a straight path that starts at one point and goes on without end in one direction. Two rays that start at the same point form an angle, and that shared point is the vertex. You name an angle with three points and put the vertex in the middle: $\angle ABC$ has its vertex at $B$ , with one side passing through $A$ and the other through $C$ . Write $m\angle ABC$ for the measure of $\angle ABC$ in degrees.

An angle bisector is a ray that starts at the vertex, runs between the two sides, and splits the angle into two angles with equal measures. Ray $BD$ is the ray that starts at $B$ and passes through $D$ . If ray $BD$ bisects $\angle ABC$ , it makes two parts, $\angle ABD$ and $\angle DBC$ , and two facts are true:

The two parts are equal: $m\angle ABD = m\angle DBC$ .
The two parts make the whole: $m\angle ABD + m\angle DBC = m\angle ABC$ .

For example, if ray $BD$ bisects $\angle ABC$ and $m\angle ABC = 90^\circ$ , then each part measures $45^\circ$ , because $45^\circ + 45^\circ = 90^\circ$ .

Both kinds of bisecting follow one pattern: two equal parts make a whole. Let the letter $p$ stand for the size of one part: a length in a unit such as centimeters, or an angle measure in degrees. A letter that stands for a number is a variable. The other part is also $p$ , so the whole is $p + p$ in the same unit.

A calculation written with numbers, variables and operations, such as $p + p$ , is an expression. Adding any number to itself gives two of that number, so $p + p$ and $2p$ are equal for every value of $p$ . Expressions are equivalent only when they are equal for every value of the variable, and these two are. That gives an equation, a statement that two expressions are equal: $\text{whole} = 2 \times \text{part}$ .

Now divide both sides of that equation by $2$ . The two sides stay equal because you do the same operation to both, and you may divide by $2$ because $2$ is not zero. The result is $\text{whole} \div 2 = \text{part}$ . The paper strip agrees: $2 \times 5 = 10$ and $10 \div 2 = 5$ , in centimeters.

Key concept

Bisect means two equal parts

When a problem states that a point is the midpoint of a segment, or that a ray bisects an angle, the whole is made of two equal parts.

If you know one part, double it to get the whole. The other part equals the part you know.
If you know the whole, halve it to get each part.

Use these rules only when the problem tells you there is a midpoint or a bisector. A point just somewhere on a segment, or a ray just somewhere inside an angle, does not have to make equal parts.

\text{whole} = 2 \times \text{part} \qquad \text{part} = \text{whole} \div 2

Four cases with stated lengths and measures

What is stated	What you know	What you do	What you find
$M$ is the midpoint of segment $AB$	The whole: $AB = 18$ cm	Halve: $18 \text{ cm} \div 2 = 9 \text{ cm}$	$AM = 9$ cm and $MB = 9$ cm
$M$ is the midpoint of segment $AB$	One part: $AM = 6.5$ cm	Double: $2 \times 6.5 \text{ cm} = 13 \text{ cm}$	$MB = 6.5$ cm and $AB = 13$ cm
Ray $BD$ bisects $\angle ABC$	The whole: $m\angle ABC = 120^\circ$	Halve: $120^\circ \div 2 = 60^\circ$	$m\angle ABD = 60^\circ$ and $m\angle DBC = 60^\circ$
Ray $BD$ bisects $\angle ABC$	One part: $m\angle DBC = 25^\circ$	Double: $2 \times 25^\circ = 50^\circ$	$m\angle ABD = 25^\circ$ and $m\angle ABC = 50^\circ$

Worked example

Halving a segment

Point $M$ is the midpoint of segment $PQ$ . Segment $PQ$ is $17$ cm long. How long is segment $PM$ ?

The 17 cm is the length of the whole segment PQ. Because M is the midpoint, PM and MQ are two equal parts that make up the whole.
$PM = MQ \quad \text{and} \quad PM + MQ = 17 \text{ cm}$
You know the whole and want one part, so halve the whole. Two equal lengths that add to 17 cm must each be 17 cm divided by 2.
$PM = 17 \text{ cm} \div 2 = 8.5 \text{ cm}$
Check with the facts from the first step: the two parts are equal, and together they rebuild the whole.
$8.5 \text{ cm} + 8.5 \text{ cm} = 17 \text{ cm}$

Answer

Segment $PM$ is $8.5$ cm long, and segment $MQ$ is $8.5$ cm long too.

Worked example

Doubling an angle

Ray $EG$ bisects $\angle DEF$ . The measure of $\angle DEG$ is $37^\circ$ . What is the measure of $\angle DEF$ ?

Read the names. The vertex is E, the middle letter, and ray EG splits angle DEF into two parts, angle DEG and angle GEF. So 37 degrees is the measure of one part. A bisector makes equal parts, so the other part has the same measure.
$m\angle GEF = m\angle DEG = 37^\circ$
You know one part and want the whole, so add the two equal parts. That is the same as doubling one part.
$m\angle DEF = 37^\circ + 37^\circ = 2 \times 37^\circ = 74^\circ$

Answer

The measure of $\angle DEF$ is $74^\circ$ .

Halving a part by mistake. Suppose ray $BD$ bisects $\angle ABC$ and $m\angle ABD = 35^\circ$ . A common mistake is to see the word bisects, think of halving, and write $m\angle ABC = 35^\circ \div 2 = 17.5^\circ$ .

This fails because $35^\circ$ is already one of the two equal parts, not the whole. The whole is made of both parts: $m\angle ABC = 35^\circ + 35^\circ = 70^\circ$ . The lengths and measures in this lesson are positive, so the whole must be greater than each part. A whole of $17.5^\circ$ would be smaller than its own $35^\circ$ part, which cannot happen.

Read the name first, then decide: is the given number a part or the whole?

Sometimes a problem gives a length or a measure as an expression with a variable in it. The meaning of bisect still gives you a fact to write down: the two parts are equal. Written out, that fact is an equation.

Keep three rules when you solve it:

Do the same operation to both sides, so the two sides stay equal.
Never divide by zero. You may divide both sides by a number only when that number is not zero.
A solution is a value of the variable that makes the original equation true, so check your value in the equation you started with.

Then use the solution to find the length or measure the problem asks for.

Worked example

Using an equation for equal parts

Ray $BD$ bisects $\angle ABC$ . The measure of $\angle ABD$ is $(3x + 5)$ degrees, where $x$ is an unknown number. The measure of $\angle DBC$ is $41$ degrees. Find $x$ , then find the measure of $\angle ABC$ .

A bisector makes two equal parts, so the measure of angle ABD equals the measure of angle DBC. Both measures are in degrees, so the expression 3x + 5 and the number 41 are equal. This is the equation to solve.
$3x + 5 = 41$
Subtract 5 from both sides so that only the term 3x is left on the left side. The two sides stay equal because you do the same operation to both.
$3x + 5 - 5 = 41 - 5 \quad \text{so} \quad 3x = 36$
Divide both sides by 3 to find x. You do this to both sides, and you may divide by 3 because 3 is not zero.
$\frac{3x}{3} = \frac{36}{3} \quad \text{so} \quad x = 12$
Check: a solution must make the original equation true. Put 12 in place of x in the equation from the first step. The left side equals 41, the same as the right side, so 12 is the solution. The measure of 41 degrees is also positive, as every angle measure in this lesson is.
$3(12) + 5 = 36 + 5 = 41$
Now find the whole. Each of the two equal parts measures 41 degrees, and the whole is double one part.
$m\angle ABC = 2 \times 41^\circ = 82^\circ$

Answer

The value of $x$ is $12$ . Each part measures $41^\circ$ , so the measure of $\angle ABC$ is $82^\circ$ .

To bisect is to make two equal parts. When a problem states that a point is the midpoint of a segment, or that a ray bisects an angle, the two parts are equal, the whole is double one part, and each part is half of the whole. First decide whether the number you were given is a part or the whole, then double or halve. When a part is given as an expression, write the equal parts as an equation, do the same operation to both sides, never divide by zero, and check that your solution makes the original equation true.