Angles
It is important to know the parts of the angles, some of which are labeled in the picture below. The sides of the angles are rays, both with the same initial point at point \(I\). The actual angle measure is not given, however the spot where the angle would be measured is shown with an arc.
It is important to know the parts of the angles, some of which are labeled in the picture below. The sides of the angles are rays, both with the same initial point at point \(I\). The actual angle measure is not given, however the spot where the angle would be measured is shown with an arc.
Measuring Angles
As shown above, an angle is made up of two rays with the same initial point. Angles are measured by how large the opening is between the rays. Angles can be measured in units called degrees or radians. We will measure angles using degrees for now, which are \(\Large\frac{1}{360}\) of a circle, by using a protractor. Use the app below to measure the angles by moving and rotating the protractor.
As shown above, an angle is made up of two rays with the same initial point. Angles are measured by how large the opening is between the rays. Angles can be measured in units called degrees or radians. We will measure angles using degrees for now, which are \(\Large\frac{1}{360}\) of a circle, by using a protractor. Use the app below to measure the angles by moving and rotating the protractor.
Classifying Angles
Angle measures help classify angles. For example, a right angle always has a measure of \(90\) degrees. If an angle is known to be a right angle, there is no need to measure the angle. It must be \(90^{\circ}\)!
Angle measures help classify angles. For example, a right angle always has a measure of \(90\) degrees. If an angle is known to be a right angle, there is no need to measure the angle. It must be \(90^{\circ}\)!
Congruent Angles
If two angles have exactly the same measure (in degrees or radians), then they are congruent.
If two angles have exactly the same measure (in degrees or radians), then they are congruent.
Since angles are made up of two rays that intersect at the vertex of the angle, the only thing we can measure is the degree or radian measure of the angle. The “length” of each side doesn’t matter since the rays continue forever in one direction. The direction of the angle does not matter either.
For example:
For example:
If \(\angle B\) and \(\angle D\) have the same angle measures, then \(\angle B\cong\angle D\)
Angle Bisectors
An angle bisector divides an angle into two congruent angles. An angle bisector is a ray, segment, or line that goes through the vertex and interior of the angle
An angle bisector divides an angle into two congruent angles. An angle bisector is a ray, segment, or line that goes through the vertex and interior of the angle
We use arc marks to show that the angles have the same measure and are congruent.
Another way to say this: If two angles are adjacent angles, then the measure of the large angle formed by their non-adjacent sides is equal to the sum of the measures of the two adjacent angles.
Example 1:
Determine the measures of the unknown angles, given that \(\overrightarrow{BD}\) bisects \(\angle{ABC}\).
Example 1:
Determine the measures of the unknown angles, given that \(\overrightarrow{BD}\) bisects \(\angle{ABC}\).
Solution
If a ray bisects an angle, then it cuts the angle into two congruent angles. Therefore, \(\angle{ABD}\cong\angle{CBD}\). Since \(m\angle{CBD}=20^{\circ}\), and since congruent angles have the same measure, then \(\angle{ABD}=20^{\circ}\) as well.
To find the larger angle, \(\angle{ABC}\), use the angle addition postulate:
\(\begin{align*}
m\angle{ABD} + m\angle{CBD} &= m\angle{ABC}\\
20^{\circ} + 20^{\circ} &=40^{\circ}\\
m\angle{ABC} &= 40^{\circ}
\end{align*}\)
Example 2
\(m\angle BAC = 53^{\circ}\)
\(m\angle CAD = 21^{\circ}\)
Find \(m\angle BAD\)
If a ray bisects an angle, then it cuts the angle into two congruent angles. Therefore, \(\angle{ABD}\cong\angle{CBD}\). Since \(m\angle{CBD}=20^{\circ}\), and since congruent angles have the same measure, then \(\angle{ABD}=20^{\circ}\) as well.
To find the larger angle, \(\angle{ABC}\), use the angle addition postulate:
\(\begin{align*}
m\angle{ABD} + m\angle{CBD} &= m\angle{ABC}\\
20^{\circ} + 20^{\circ} &=40^{\circ}\\
m\angle{ABC} &= 40^{\circ}
\end{align*}\)
Example 2
\(m\angle BAC = 53^{\circ}\)
\(m\angle CAD = 21^{\circ}\)
Find \(m\angle BAD\)
Solution
Since \(\angle{BAC}\) and \(\angle{CAD}\) share a vertex (\(A\)) and a ray (\(\overrightarrow{AC}\)), they are adjacent angles. Therefore, using the angle addition postulate, their measures add up to the large angle, \(\angle {BAD}\).
\(\begin{align*}
m\angle BAC + m\angle CAD &= m\angle BAD\\
53^{\circ} + 21^{\circ} &= m\angle BAD\\
74^{\circ} &= m\angle BAD
\end{align*}\)
Example 3
Given that two adjacent angles add up to \(139^{\circ}\) and that one of the angles is \(55^{\circ}\), what is the measure of the other angle?
Solution
Draw a picture to show the situation:
\(m\angle CLN = 55^{\circ}\)
\(m\angle RLN = 139^{\circ}\)
Find \(m\angle RLC\)
Since \(\angle{BAC}\) and \(\angle{CAD}\) share a vertex (\(A\)) and a ray (\(\overrightarrow{AC}\)), they are adjacent angles. Therefore, using the angle addition postulate, their measures add up to the large angle, \(\angle {BAD}\).
\(\begin{align*}
m\angle BAC + m\angle CAD &= m\angle BAD\\
53^{\circ} + 21^{\circ} &= m\angle BAD\\
74^{\circ} &= m\angle BAD
\end{align*}\)
Example 3
Given that two adjacent angles add up to \(139^{\circ}\) and that one of the angles is \(55^{\circ}\), what is the measure of the other angle?
Solution
Draw a picture to show the situation:
\(m\angle CLN = 55^{\circ}\)
\(m\angle RLN = 139^{\circ}\)
Find \(m\angle RLC\)
Using the angle addition postulate, we know that the two adjacent angles must add up to the larger angle. So
\(\begin{align*}
m\angle{RLC} + m\angle{CLN} &= m\angle{RLN}\\
m\angle{RLC} + 55^{\circ} &=139^{\circ}\\
m\angle{RLC} &= 84^{\circ}
\end{align*}\)
\(\begin{align*}
m\angle{RLC} + m\angle{CLN} &= m\angle{RLN}\\
m\angle{RLC} + 55^{\circ} &=139^{\circ}\\
m\angle{RLC} &= 84^{\circ}
\end{align*}\)
Quick Check
1) Name the vertex and sides of each angle:
1) Name the vertex and sides of each angle:
2) Name each angle above in three different ways. Then classify the angle.
3) Find the missing angle measures.
3) Find the missing angle measures.