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What's an Angle?
An angle (represented by the symbol \(\angle\) ) is formed by two rays which share the same endpoint.
The shared endpoint is called the vertex of the angle.
Naming Angles
There are three ways to name an angle:
- Name the vertex: \(\angle A\)
- Name three points, with the vertex in the middle: \(\angle CAB\) or \(\angle BAC\)
- Name the number inside the angle: \(\angle 1\)
Naming With Multiple Angles
If multiple angles share the same vertex, you cannot name them using only the vertex. You must name them with their numbers, or with three points in the angle.
In this example, naming it \(\angle A\) would be inappropriate, because we wouldn't know if it was referring to angle 3 or 4.
The space between the rays of an angle is the interior of the angle, the surrounding space is the exterior.
Angles can be measured on paper using a protractor. There are many types of protractors with varying levels of accuracy. The most common type of protractor used in educational settings is a half circle protractor, usually made of plastic. It looks like this:
2.2: Measuring Angles
Angles are measured using degrees, which are based on a 360° circle. The little circle ° means degrees.
In the circle shown, there are four different angles being displayed. The first, innermost angle, in the little green area, is a 90° angle, and goes 1/4 of the way around the circle.
The next angle, in shaded in dark greyish blue, is 180°, and goes halfway around the circle.
The white shaded angle, which goes 3/4 of the way around, is 270° and finally, the full circle is 360 degrees. 360° is a full rotation.
You can show that an angle is being measured by placing an m in front of the angle symbol. So \(m\angle A\) means "the measure of angle A."
2.3: Types of Angles
There are several terms related to categorizing angles which you should familiarize yourself with.
Right Angles
A right angle is an angle that measures exactly ninety degrees. It's indicated by drawing a right angle symbol (shown in picture) in the interior of the angle.
Acute Angles
An acute angle is an angle that measures anywhere between 0 and 90 degrees (but not 0 or 90 degrees exactly).
An easy way to remember this is "a cute angle." Cute things are usually little, and acute angles are also little.
Obtuse Angles
An obtuse angle is an angle that measures anywhere between 90 and 180 degrees (but not 90 or 180 degrees exactly).
Straight Angles
A straight angle is an angle that measures exactly 180 degrees. It's a straight line.
Adjacent Angles
Two angles are said to be adjacent to one another if they share a side and vertex, and no common interior points. In this example, angle 1 is adjacent to angle 2. They are right next to each other, but don't overlap.
Non-Adjacent Angles
Angles are not adjacent if they don't share a side or a vertex. In this example, \(\angle 1\) is not adjacent to \(\angle 2\) because they aren't sharing a common side. Angle 3 is not adjacent to angle 4 because they don't have a shared side or shared vertex.
Congruent Angles
Congruent angles are angles that have the same measure. Congruence can be shown by matching angle marks, similar to tick marks for matching congruent line segments. The congruency symbol \(\cong\) may also be used.
In the above example, both 55° angles are congruent, and \(\angle Q \cong \angle R\), since they have matching angle marks.
Complementary Angles
Complementary angles are angles whose measures add up to 90º - They form a right angle when combined.
Note that they don't have to be together to be complementary. They just have to add up to ninety degrees. Angles A and B in the above example are complementary, even though they're not connected.
Supplementary Angles
Supplementary angles are angles whose measures add up to 180º
Note that they don't have to be together to be supplementary. They just have to add up to 180 degrees. Angles A, B, and C in the above example are supplementary, even though they're not connected.
Linear Pair
Two adjacent angles (they must share a side!) that are also supplementary form a linear pair. Linear, meaning line, and pair, meaning two.
Vertical Angles
Vertical angles are angles that are opposite each other when two lines intersect.
Vertical angles are congruent.
In the above example, 5 is not a vertical angle to 7 and 6 is not vertical to 8 because there's only one line present.
2.4: Angle Addition Postulate
The Angle Addition Postulate states that when two angles are adjacent, the resulting angle will be the sum of the adjacent angles.
Although this might appear obvious, it's important to know. We can use this concept to figure out missing information, given a place to start.
Example Problems
Example 1
Instructions: Find the measure of \(\angle ABC\) using the figure above.
We know this is a supplementary angle, because \(\overleftrightarrow{AD}\) is a straight line. These two angles form a linear pair, so the sum of their angles is 180°.
\(m\angle ABD = 180°\) because \(\overleftrightarrow{AD}\) is a straight line
\(m\angle ABD = 35° + m\angle ABC\) because of the Angle Addition Postulate
Solve for \(m\angle ABC\)
\(m\angle ABC = 180° - 35°\)
\(m\angle ABC = 145°\)
Example 2
Instructions: Find \(m\angle DBE\)
This example has the two straight lines \(\overleftrightarrow{AC}\) and \(\overleftrightarrow{FE}\) in it. They intersect at point B, with another ray \(\overrightarrow{BD}\) extending off it.
\(m\angle FBC = 135°\) and it is a vertical angle to \(\angle ABE\)
Since \(\angle FBC\) and \(\angle ABE\) are verticals, we know that they must be congruent. \(\angle FBC \cong \angle ABE\). We now know \(m\angle ABE = 135°\)
By the Angle Addition Postulate, \(m\angle ABE = m\angle ABD + m\angle DBE\)
Substitute and solve. \(135° = 55° + m\angle DBE\)
\(m\angle DBE=135°-55°\)
\(m\angle DBE=80°\)
Example 3
Instructions: Solve for x.
The angles in this example do not have labels, but we can still solve for x, because we know there are two straight lines intersecting. Since the angle containing \(4x-2\) is supplementary to the 130° angle, we know that \(4x-2=50°\)
Now all we need to do is solve for x.
\(4x=50+2\)
\(4x=52\)
\(x=52\div{4}\)
\(x=13\)
Example 4
Instructions: Angles 1 and 2 make a linear pair. Given \(m\angle{1}=64°\), find \(m\angle{2}\).
A linear pair is two angles that add up to 180° - So we just need to subtract 64° from 180° to find what \(m\angle{2}\) is.
\(m\angle{2} = 180°-64°\)
\(m\angle{2} = 116°\)
You can draw it if you'd like to visualize what's happening:
Angle Bisectors
An angle bisector is a ray that divides an angle into two congruent parts. It splits it down the middle.
In this example, ray CB bisects angle ACD. Therefore, ACB is congruent to BCD.
Perpendicular Bisector
A perpendicular bisector intersects a segment at a right angle (90°), splitting it into two congruent parts. A ray, line, or another segment can be a perpendicular bisector.
In these examples, \(\overrightarrow{CD}\) is a perpendicular bisector of \(\overline{AB}\), line l is a perpendicular bisector of \(\overline{AB}\), and \(\overline{EF}\) is a perpendicular bisector of \(\overline{AB}\).
Angles: Review Questions
1. Visualize: Draw a right angle \(\angle{ABC}\). Draw \(\overrightarrow{BD}\) so it bisects \(\angle{ABC}\). What is \(m\angle{ABD}\)?
2. Calculate: Determine \(m\angle{OBP}\) using the figure below:
3. Visualize: Point B is on \(\overline{AC}\). \(\overrightarrow{BD}\) is a perpendicular bisector and extends out above of \(\overline{AC}\). AB=12in. Draw this figure. What is the length of \(\overline{BC}\)?
4. Calculate: \(\angle{1}\) and \(\angle{2}\) form a linear pair. \(m\angle{1}=120°\). What is \(m\angle{2}\)?
5. Definition: Are \(\angle{ABR}\) and \(\angle{QBC}\) vertical angles? Why or why not?
6. Calculate: Determine \(m\angle{DBC}\) using the given figure.
7. Calculate: Determine the value of x.
8. Calculate: Determine the measures of all the angles in this figure.
1. \(m\angle{ABD}\) must be 45° because \(\overrightarrow{BD}\) bisects \(\angle{ABC}\). Since \(\angle{ABC}\) is a right angle, we know it's 90° and half of 90° is 45°
The figure looks like this:
2. Since \(\overleftrightarrow{AC}\) is a line, we know \(m\angle{ABC}=180°\). With the angle addition postulate, we know \(m\angle{ABC}=m\angle{ABO}+m\angle{OBP}+m\angle{PBC}\).
When we substitute, \(180°=26.5°+m\angle{OBP}+45°\).
\(m\angle{OBP}=180°-26.5°-45°\)
\(m\angle{OBP}=108.5°\)
3. Since \(\overline{AB}\) is 12 inches long, and \(\overrightarrow{BD}\) is a perpendicular bisector of \(\overline{AC}\), we know \(\overline{BC}\) must also be 12 inches long.
It should look something like this:
4. Because these angles form a linear pair, we know their total measure is 180 degrees. \(m\angle{2}\) must be 60 degrees.
5. While these are opposite angles, they are not vertical angles. To be vertical angles, there must be two lines present. There is only one line in this example. The angles are not congruent.
6. \(m\angle{DBC}\) forms a linear pair with \(m\angle{CBR}\). This means \(m\angle{DBC}\) and \(m\angle{CBR}\) combine to form a 180° angle. \(m\angle{DBC}=135°\) because \(180°-45°=135°\)
7. \(\angle{DBA}\) and \(\angle{CBR}\) are vertical angles. This means \(\angle{DBA}\cong\angle{CBR}\), so \(m\angle{DBA}=60°\).
Now it's a matter of solving the equation for x.
\(4x-20=60\)
\(4x=60+20\)
\(4x=80\)
\(x=80\div 4\)
\(x=20\)8. \(\angle{DBA}\) and \(\angle{DBC}\) form a linear pair, so \(m\angle{DBA}+m\angle{DBC}=180°\)
Substitute
\(4x+(20x+60)=180\)
Solve
\(24x+60=180\)
\(24x=120\)
\(x=5\)
Plug the value of x into the equations to find \(m\angle{DBA}\)
\(m\angle{DBA}=4\times 5=20°\)
You can plug in x to find \(m\angle{DBC}\) or use the properties of a linear pair to know \(m\angle{DBC}=160°\)
With vertical angles or linear pairs, you can determine the measures of the remaining angles.
