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6. 2D Geometry
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United States of AmericaVirginia
Grade 8

6.01 Angle relationships

Lesson
Practice

Angle relationships

Whenever two non-overlapping angles share a common ray and a common vertex, we say they are adjacent angles. Here are some examples:

Two sets of adjacent angles.

Vertical angles are a pair of nonadjacent angles formed by two intersecting lines. Vertical angles are congruent and share a common vertex.

3 images. Each showing vertical angles formed by 2 intersecting lines. Ask your teacher for more information.

Complementary angles are any two angles such that the sum of their measures is 90\degree.

Supplementary angles are any two angles such that the sum of their measures is 180\degree.

An image showing 2 complementary angles forming a right angle and 2 supplementary angles forming a straight angle.
Two rays intersecting at point A. The four angles formed are labeled 1 to 4 from upper right clockwise.

\,\\\,\\\,\\\,Previously, we would name angles using a single letter at the vertex.

Notice in this image, \angle A could refer to four different angles since they all share the same vertex point A. We would need to refer to these angles by their number instead.

Line BD intersecting line CE at point A. The four angles created are marked 1 to 4.

\,\\\,\\\,\\\,If we add an additional point on each ray from the vertex, we can now refer to each angle distintly without using numbers.

  • \angle 1 is equivalent to \angle CAD

  • \angle 2 is equivalent to \angle BAC

  • \angle 3 is equivalent to \angle EAB

  • \angle 4 is equivalent to \angle DAE

We can name angles in two different ways as long as the vertex is the middle letter:

Two equal angles. The first is named angle B A C and second is named angle C A B. Ask your teacher for more information.

We can use these definitions and relationships to solve for unknown measures. In the next image, we can see that \angle BAC and \angle CAD are supplementary since they form a straight angle. Since supplementary angles sum to 180 \degree, we know m\angle CAD = 180 \degree - 115 \degree = 65 \degree.

line BD with point A somewhere in the middle. Points C and A are connected by a line. The angle formed, BAC, is 115 degrees in measurement.
Angle BAD, with ray AC diving the angle into two, angle BAC measures 65 degrees.

\,\\\,\\\,\\\,If we know the measure of \angle BAD, and either of the smaller adjacent angles, we can find the value of the other angle.

m\angle BAD = m\angle BAC + m\angle CAD

m\angle CAD = m\angle BAD - m\angle BAC

If m\angle BAD = 80 \degree:m\angle CAD = 80 \degree - 65 \degree = 15 \degree

Examples

Example 1

Which of these diagrams shows a pair of adjacent angles?

A
Two segments intersecting with two marked angles lie on the opposite sides.
B
Two segments intersecting with two marked angles lie on the opposite sides.
C
A pair of adjacent angles.
D
Two intersecting lines where a pair of vertical angles are formed.
Worked Solution
Create a strategy

We should look for two non-overlapping angles that share a common ray and a common vertex.

Apply the idea

The option that shows the two angles touching along a common segment is option C.

A pair of adjacent angles.

Example 2

Name an angle that is supplementary with \angle 3 in the figure below:

Two intersecting lines. The four angles formed are labeled 1 to 4 from left clockwise.

Use the angle symbol \angle in your answer.

Worked Solution
Create a strategy

Supplementary angles are two angles forming a straight angle.

Apply the idea

Both \angle 2 and \angle 4 are supplemetary to \angle 3.

Reflect and check

Notice the supplementary angles are adjacent to \angle 3.

If we wanted to find the angles congruent to \angle 3, it would be \angle 1 since they are vertical angles.

Example 3

The angles in the diagram below are complementary. What is the value of x?

A right angle formed by complementary angles measuring 39 degrees and x degrees.
Worked Solution
Create a strategy

Complementary angles are two angles forming a right angle equivalent to 90\degree.

Apply the idea
\displaystyle x + 39\displaystyle =\displaystyle 90Equate the sum of the angles to 90
\displaystyle x\displaystyle =\displaystyle 51Subtract 39 from both sides

Example 4

If the measure of angle \angle BAD \text{ is } 5x-4 \degree, find the measure of \angle CAD.

Angles BAC and CAD. Angle BAC is labeled '3x+1 degrees'.
Worked Solution
Create a strategy

Since \angle BAC and \angle CAD are adjacent:m\angle BAD = m\angle BAC + m\angle CAD

Apply the idea
\displaystyle m\angle BAC + m\angle CAD\displaystyle =\displaystyle m\angle BAD
\displaystyle 3x-1 \degree + m\angle CAD\displaystyle =\displaystyle 5x-4 \degreeSubstitute the value of each angle
\displaystyle m\angle CAD\displaystyle =\displaystyle 5x-4 \degree - \left(3x-1 \degree\right)Isolate the unknown angle
\displaystyle m\angle CAD\displaystyle =\displaystyle 2x-3 \degreeSubtract the expressions
Idea summary

Adjacent angles are two non-overlapping angles that share a common ray and a common vertex.

Vertical angles are a pair of nonadjacent angles formed by two intersecting lines. Vertical angles are congruent and share a common vertex.

Complementary angles are angles whose sum is 90 \degree.

Supplementary angles are angles whose sum is 180 \degree.

Outcomes

8.MG.1

The student will use the relationships among pairs of angles that are vertical angles, adjacent angles, supplementary angles, and complementary angles to determine the measure of unknown angles.

8.MG.1a

Identify and describe the relationship between pairs of angles that are vertical, adjacent, supplementary, and complementary.

8.MG.1b

Use the relationships among supplementary, complementary, vertical, and adjacent angles to solve problems, including those in context, involving the measure of unknown angles.

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