Vertical and Adjacent Angles: Find and Solve | Grade 7 Math | LevelKite

You measure angles in degrees ( $^\circ$ ). In the last lesson you used two facts. Complementary angles add to $90^\circ$ . Supplementary angles add to $180^\circ$ . Now you will use them to find angles that sit next to each other.

Adjacent angles share a vertex and one side, and they do not overlap.
A linear pair is two adjacent angles whose other two sides form one straight line. Together they make a straight angle, so they are supplementary and add to $180^\circ$ .
Two adjacent angles that together fill a right angle are complementary and add to $90^\circ$ . A small square mark shows a right angle.

You write $m\angle AEC$ for "the measure of angle AEC".

When two lines cross, they make four angles at the crossing point. Vertical angles are the two angles directly across from each other. They share only the vertex, so they are not adjacent.

In the figure below, lines AB and CD cross at point E. Angle AEC and angle BED are vertical angles. Here is why they are always equal:

Angle AEC and angle CEB form a linear pair along line AB. So $m\angle AEC = 180^\circ - m\angle CEB$ .
Angle CEB and angle BED form a linear pair along line CD. So $m\angle BED = 180^\circ - m\angle CEB$ .

Both angles equal $180^\circ$ minus the same angle, so they are equal. This reasoning works for any two lines that cross.

Lines AB and CD cross at E. In Example 1, angle AEC measures

50^\circ

. The figure is not to scale, so use only this stated value.

Worked example

Example 1: Angles at two crossing lines

In the figure above, lines AB and CD cross at point E, and angle AEC measures $50^\circ$ . Find the measures of angle BED, angle CEB and angle AED.

Read the figure first. Angle BED sits directly across point E from angle AEC, so they are vertical angles. Vertical angles are equal.
$m\angle BED = m\angle AEC = 50^\circ$
Angle CEB shares ray EC with angle AEC, and rays EA and EB form line AB. So the two angles are a linear pair and add to $180^\circ$ . Subtract the known angle from $180^\circ$ .
$m\angle CEB = 180^\circ - 50^\circ = 130^\circ$
Angle AED sits directly across point E from angle CEB, so they are vertical angles and are equal.
$m\angle AED = m\angle CEB = 130^\circ$
Check. Each pair of angles next to each other is a linear pair: AEC with CEB, CEB with BED, BED with AED, and AED with AEC. Each pair must add to $180^\circ$ , and each pair is one $50^\circ$ angle and one $130^\circ$ angle.
$50^\circ + 130^\circ = 180^\circ$

Answer

Angle BED measures $50^\circ$ , and angle CEB and angle AED each measure $130^\circ$ .

Key concept

Name the relationship, then calculate

Before you compute, read how the angles sit in the figure:

Vertical angles (directly across from each other where two lines cross) are equal.
A linear pair (adjacent angles whose other sides form one straight line) adds to $180^\circ$ .
Adjacent angles that together fill a right angle (square mark) add to $90^\circ$ .

Use only the marks and the values stated in the text, never how big an angle looks in the drawing.

Sometimes an angle's measure is given as an expression. A variable, such as $x$ , stands for an unknown number. The expression $(3x + 10)^\circ$ gives the angle's measure in degrees once you know $x$ .

To find an unknown angle, follow these steps:

Read the relationship from the figure.
Write an equation. Set the expressions equal for vertical angles. Set their sum to $180$ for a linear pair. Set their sum to $90$ for adjacent angles inside a right angle.
Solve in steps. You may replace an expression with an equivalent one, which is equal for every value of $x$ , such as $5x + 2x = 7x$ . Do the same operation to both sides so they stay equal, and never divide by zero.
Substitute $x$ into each expression to find each angle.
Check that each angle is positive, because an angle measure must be greater than $0^\circ$ . Also check that the relationship holds. A solution must make the original equation true.

The square mark at B makes angle ABC a right angle. In Example 2, angle ABD measures

(5x - 12)^\circ

and angle DBC measures

(2x + 4)^\circ

.

Worked example

Example 2: An equation inside a right angle

In the figure above, the square mark shows that angle ABC is a right angle, and ray BD lies inside it. Angle ABD measures $(5x - 12)^\circ$ and angle DBC measures $(2x + 4)^\circ$ . Find $x$ and the measure of each angle.

Read the figure first. The square mark means angle ABC measures $90^\circ$ . Ray BD splits it into two adjacent angles, angle ABD and angle DBC. Together they fill the right angle, so they add to $90^\circ$ .
$m\angle ABD + m\angle DBC = 90^\circ$
Write the equation by replacing each measure with its expression. Then combine like terms: $5x + 2x = 7x$ and $-12 + 4 = -8$ . The new expression is equivalent, because it is equal for every value of $x$ .
$(5x - 12) + (2x + 4) = 90 \;\Rightarrow\; 7x - 8 = 90$
Add 8 to both sides. Doing the same operation to both sides keeps the two sides equal.
$7x - 8 + 8 = 90 + 8 \;\Rightarrow\; 7x = 98$
Divide both sides by 7. This is allowed because 7 is not zero.
$x = 98 \div 7 = 14$
Substitute $x = 14$ into each original expression to find each angle.
$m\angle ABD = 5(14) - 12 = 70 - 12 = 58^\circ, \quad m\angle DBC = 2(14) + 4 = 28 + 4 = 32^\circ$
Check. Both measures are positive, and they add to $90^\circ$ . So the relationship holds, and $x = 14$ makes the original equation true.
$58^\circ + 32^\circ = 90^\circ$

Answer

$x = 14$ , so angle ABD measures $58^\circ$ and angle DBC measures $32^\circ$ .

Lines AB and CD cross at E. In Example 3, angle AEC measures

(3x + 10)^\circ

and angle BED measures

(5x - 30)^\circ

. The figure is not to scale.

Worked example

Example 3: An equation with vertical angles

In the figure above, lines AB and CD cross at point E. Angle AEC measures $(3x + 10)^\circ$ and angle BED measures $(5x - 30)^\circ$ . Find $x$ , the measures of angle AEC and angle BED, and the measure of angle CEB.

Read the figure. Angle AEC and angle BED are directly across point E from each other, so they are vertical angles and are equal. Set the two expressions equal.
$3x + 10 = 5x - 30$
Subtract $3x$ from both sides to collect the $x$ terms on one side. Then add 30 to both sides. Each operation is done to both sides, so the sides stay equal.
$10 = 2x - 30 \;\Rightarrow\; 40 = 2x$
Divide both sides by 2, which is not zero.
$x = 40 \div 2 = 20$
Substitute $x = 20$ into each original expression. Both angles are positive and equal. So the vertical-angle relationship holds, and $x = 20$ makes the original equation true.
$3(20) + 10 = 60 + 10 = 70, \quad 5(20) - 30 = 100 - 30 = 70$
Angle CEB shares ray EC with angle AEC, and rays EA and EB form line AB. So the two angles are a linear pair and add to $180^\circ$ . Check: $70^\circ + 110^\circ = 180^\circ$ .
$m\angle CEB = 180^\circ - 70^\circ = 110^\circ$

Answer

$x = 20$ . Angle AEC and angle BED each measure $70^\circ$ , and angle CEB measures $110^\circ$ .

Takeaway: Name the relationship before you calculate. Vertical angles are equal. A linear pair adds to $180^\circ$ . Adjacent angles that fill a right angle add to $90^\circ$ . When angle measures are given as expressions in $x$ , that relationship becomes your equation. Solve it by doing the same operation to both sides. Then substitute and check that each angle is positive and that the relationship holds.

Next, you will use these relationships together to solve multi-step problems in one figure. You will also find and fix errors in angle reasoning.