6.4: Measures of Angle Pairs
Candace opened her math book and discovered this dilemma.
The two angles below are complementary. \begin{align*}m\angle GHI = x\end{align*}. What is \begin{align*}x\end{align*}?
Candace is puzzled. Are you? Using equations and geometry can help you figure this one out. Pay attention and you will learn all that you need to know from this Concept.
Guidance
Let's think about angle pairs. There are different types of angle pairs.
- Supplementary angles are two angles that form a straight line, and their sum is always \begin{align*}180^{\circ}\end{align*}.
- Complementary angles together form a right angle and have a sum of \begin{align*}90^{\circ}\end{align*}.
- Adjacent angles are next to each other. When they form a line, their sum is \begin{align*}180^{\circ}\end{align*}.
- Vertical angles are directly opposite each other. They are equal.
Take a look at this situation involving angle pairs.
Fill in the figure below with the angle measures for all of the angles shown.
First, notice that we only have one angle to go on. This angle measures 70 degrees. However, that is enough information to figure out all of the other angles in this diagram. We can use the information that we know about angles to figure the measures of these angles out.
Let’s begin with adjacent angles. Angle \begin{align*}b\end{align*} is adjacent to the 70 degree angle. Since we know that adjacent angles form a straight line, the sum of the two angles is \begin{align*}180^{\circ}\end{align*}.
We can write this equation.
\begin{align*}180 = 70 + b\end{align*}
We know that \begin{align*}b\end{align*} is equal to \begin{align*}110^{\circ}\end{align*}.
Next, we can work on the vertical angles. Angle \begin{align*}c\end{align*} is vertical with angle \begin{align*}b\end{align*}. Vertical angles have the same measure, so the measure of angle \begin{align*}c\end{align*} is also \begin{align*}110^{\circ}\end{align*}.
Angle \begin{align*}a\end{align*} is vertical with the \begin{align*}70^{\circ}\end{align*} given angle, we know that this one is also \begin{align*}70^{\circ}\end{align*}.
Using our known information, we have figured out the measures of all of the missing angles.
Find the measure of an angle that forms a supplementary angle with \begin{align*}\angle MRS\end{align*} if \begin{align*} m\angle MRS\end{align*} is:
Example A
\begin{align*}61^\circ\end{align*}
Solution: \begin{align*}119^\circ\end{align*}
Example B
\begin{align*}40^\circ\end{align*}
Solution: \begin{align*}140^\circ\end{align*}
Example C
\begin{align*}121^\circ\end{align*}
Solution: \begin{align*}59^\circ\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
If the two angles are complementary, then we know that the sum of the two angles is 90 degrees.
Let's write an equation to show that.
\begin{align*}x + 34 = 90\end{align*}
Now we can solve for \begin{align*}x\end{align*}.
\begin{align*}x = 90 - 34\end{align*}
\begin{align*}x = 56^\circ\end{align*}
This is the answer.
Vocabulary
- Parallel lines
- lines that are an equal distance apart and will never intersect.
- Intersecting lines
- lines that cross at one point.
- Perpendicular lines
- lines that intersect at a \begin{align*}90^{\circ}\end{align*} angle and form two or more \begin{align*}90^{\circ}\end{align*} angles.
- Angle
- the measure of the space formed by two intersecting lines.
- Straight angle
- is a straight line equal to \begin{align*}180^{\circ}\end{align*}.
- Angle Pairs
- the relationship formed by two angles.
- Complementary Angles
- two angles whose sum is \begin{align*}90^{\circ}\end{align*}.
- Supplementary Angles
- two angles whose sum is \begin{align*}180^{\circ}\end{align*}.
- Adjacent Angles
- angles that are next to each other and whose sum is \begin{align*}180^{\circ}\end{align*}.
- Vertical Angles
- angles that are diagonally across from each other and whose sum is \begin{align*}90^{\circ}\end{align*}.
Guided Practice
Here is one for you to try on your own.
A pair of angles are adjacent and complementary. What is the missing measure if \begin{align*}\angle 1\end{align*} is equal to \begin{align*}x+5\end{align*}? Can you figure out the measure of \begin{align*}\angle 2\end{align*}?
Solution
To figure this out, you have to know a few things. First, adjacent angles are next to each other and complementary angles have a sum of \begin{align*}90^\circ\end{align*}.
Knowing this, we can write an equation.
\begin{align*}x+5=90\end{align*}
Now we can solve for \begin{align*}x\end{align*}
\begin{align*}x = 85\end{align*}
The measure of \begin{align*}\angle 2\end{align*} is \begin{align*}85^\circ\end{align*}.
Video Review
Complementary, Supplementary and Vertical Angles
Practice
Directions: If the following angle pairs are complementary, then what is the measure of the missing angle?
- \begin{align*}\angle{A}&=45^{\circ}\\ \angle{B}&= ?\end{align*}
- \begin{align*}\angle{C}&=83^{\circ}\\ \angle{D}&= ?\end{align*}
- \begin{align*}\angle{E}&=33^{\circ}\\ \angle{F}&= ?\end{align*}
- \begin{align*}\angle{G}&=53^{\circ}\\ \angle{H}&= ?\end{align*}
Directions: If the following angle pairs are supplementary, then what is the measure of the missing angle?
- \begin{align*}\angle{A}&=40^{\circ}\\ \angle{B}&= ?\end{align*}
- \begin{align*}\angle{A}&=75^{\circ}\\ \angle{B}&= ?\end{align*}
- \begin{align*}\angle{C}&=110^{\circ}\\ \angle{F}&= ?\end{align*}
- \begin{align*}\angle{D}&=125^{\circ}\\ \angle{E}&= ?\end{align*}
- \begin{align*}\angle{M}&=10^{\circ}\\ \angle{N}&= ?\end{align*}
- \begin{align*}\angle{O}&=157^{\circ}\\ \angle{P}&= ?\end{align*}
Directions: Define the following types of angle pairs.
- Vertical angles
- Adjacent angles
- Complementary angles
- Supplementary angles
- Interior angles
Adjacent Angles
Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.Angle
A geometric figure formed by two rays that connect at a single point or vertex.Intersecting lines
Intersecting lines are lines that cross or meet at some point.Parallel
Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.Perpendicular lines
Perpendicular lines are lines that intersect at a angle.Straight angle
A straight angle is a straight line equal to .Image Attributions
Here you'll find the measures of angle pairs by using relationships and given information.
Concept Nodes:
Adjacent Angles
Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.Angle
A geometric figure formed by two rays that connect at a single point or vertex.Intersecting lines
Intersecting lines are lines that cross or meet at some point.Parallel
Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.Perpendicular lines
Perpendicular lines are lines that intersect at a angle.Straight angle
A straight angle is a straight line equal to .