Topics
8. Foundations of Geometry
topic badge
United States of AmericaTennessee
Integrated Math 1

8.04 Angle relationships

Lesson
Practice
Lesson

Concept summary

Certain pairs of angles can have a special relationship:

  • Complementary angles have a sum of 90\degree
  • Supplementary angles have a sum of 180\degree
Linear pair

Adjacent angles that form a straight line.

A straight horizontal line with a ray extending from a point on the line. The ray creates two angles.

The linear pair postulate states that if two angles form a linear pair, then they are supplementary.

Vertical angles

The opposite angles formed when two lines intersect.

Two lines intersecting and creating two pairs of angles. The pair of angles that are across from each other and have the larger measure are marked with one congruent marking, the pair of angles with the smaller measure that are across from each other are marked with two congruent markings.

The vertical angles theorem states that vertical angles are congruent.

Perpendicular lines

Two lines that intersect at right angles. Lines are denoted as being perpendicular by the symbol \perp.

Two intersecting lines with a right angle marking.

Worked examples

Example 1

The angles in the diagram are complementary. Find the value of x.

A right angle with a point in the interior of the angle. A segment is drawn from the vertex of the angle to the point, forming two adjacent angles. The measure of the two angles are x degrees, and 39 degrees.

Approach

Complementary angles have a sum of 90\degree. Use this to write an equation that includes the two angles in the diagram knowing that the angles are complementary. Then we want to solve for x.

Solution

\displaystyle x+39\displaystyle =\displaystyle 90Definition of complementary
\displaystyle x\displaystyle =\displaystyle 51Subtraction property of equality

x=51 \degree

Reflection

If the angles were supplementary, then the sum of the angles would equal 180 \degree instead of 90 \degree.

Example 2

Use the diagram to identify an example of each angle pair.

Lines A D and E C intersecting at point F. Ray F B is in the interior of angle A F C. Angle B F C has a measure of 28 degrees, angle A F E has a measure of 62 degrees, and angle E F D has a measure of 118 degrees.
a

Vertical angles

Approach

Vertical angles are formed by intersecting lines. There is only one pair of intersecting lines in the diagram, \overleftrightarrow{AD} and \overleftrightarrow{CE}. Identify a pair of opposite angles formed by this intersection.

Solution

\angle{AFC} and \angle{EFD} or \angle{AFE} and \angle{CFD}

b

Linear pair

Approach

Linear pairs are adjacent angles that form a line. First, we need to identify a line such as \overleftrightarrow{AD} or \overleftrightarrow{CE} and see if we can identify adjacent angles that form this line.

Solution

\angle{AFE} and \angle{EFD} form a linear pair.

Reflection

There are multiple linear pairs in the diagram.

Example 3

In the diagram below, lines \overleftrightarrow{AB} and \overleftrightarrow{CD} intersect at point E.

Lines AB and CD intersect at point E. Angle AEC has a measure of 58 degrees, angle BED has a measure of 3x - 2 degrees, and angle BEC has a measure of 12 + 5y degrees.

Use the diagram to solve for each variable.

a

x

Approach

We know the measure of \angle{AEC}. We can connect this to the measure of \angle{BED}, since they are vertical angles, to form an equation involving x.

Solution

\displaystyle m\angle{AEC}\displaystyle =\displaystyle m\angle{BED}Vertical angles have equal measures
\displaystyle 58\displaystyle =\displaystyle 3x - 2Substitution property of equality
\displaystyle 60\displaystyle =\displaystyle 3xAddition property of equality
\displaystyle 20\displaystyle =\displaystyle xDivision property of equality

So we have that x = 20.

b

y

Approach

We know the measure of \angle{AEC}. We can connect this to the measure of \angle{BEC}, since they are a linear pair, to form an equation involving y.

Solution

\displaystyle m\angle{AEC} + m\angle{BEC}\displaystyle =\displaystyle 180\degreeAngle measures of a linear pair sum to 180\degree
\displaystyle 58 + 12 + 5y\displaystyle =\displaystyle 180Substitution property of equality
\displaystyle 70 + 5y\displaystyle =\displaystyle 180Simplify
\displaystyle 5y\displaystyle =\displaystyle 110Subtraction property of equality
\displaystyle y\displaystyle =\displaystyle 22Division property of equality

So we have that y = 22.

Outcomes

M1.G.CO.B.3

Use definitions and theorems about lines and angles to solve problems and to justify relationships in geometric figures.

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

M1.MP8

Look for and express regularity in repeated reasoning.

What is Mathspace

About Mathspace