1.3: Angles and Measurement
Learning Objectives
- Classify angles.
- Apply the Protractor Postulate and the Angle Addition Postulate.
Review Queue
- Label the following geometric figure. What is it called?
- Find \begin{align*}a, XY\end{align*} and \begin{align*}YZ\end{align*}.
- \begin{align*}B\end{align*} is between \begin{align*}A\end{align*} and \begin{align*}C\end{align*} on \begin{align*}\overline{AC}\end{align*}. If \begin{align*}AB = 4\end{align*} and \begin{align*}BC = 9\end{align*}, what is \begin{align*}AC\end{align*}?
Know What? Back to the building blocks. Every block has its own dimensions, angles and measurements. Using a protractor, find the measure of the three outlined angles in the “castle” to the right.
Two Rays = One Angle
In #1 above, the figure was a ray. It is labeled \begin{align*}\overrightarrow{AB}\end{align*}, with the arrow over the point that is NOT the endpoint. When two rays have the same endpoint, an angle is created.
Angle: When two rays have the same endpoint.
Vertex: The common endpoint of the two rays that form an angle.
Sides: The two rays that form an angle.
Label It | Say It |
---|---|
\begin{align*}\angle ABC\end{align*} | Angle \begin{align*}ABC\end{align*} |
\begin{align*}\angle CBA\end{align*} | Angle \begin{align*}CBA\end{align*} |
The vertex is \begin{align*}B\end{align*} and the sides are \begin{align*}\overrightarrow{BA}\end{align*} and \begin{align*}\overrightarrow{BC}\end{align*}. Always use three letters to name an angle, \begin{align*}\angle\end{align*} SIDE-VERTEX-SIDE.
Example 1: How many angles are in the picture below? Label each one.
Solution: There are three angles with vertex \begin{align*}U\end{align*}. It might be easier to see them all if we separate them.
So, the three angles can be labeled, \begin{align*}\angle XUY\end{align*} (or \begin{align*}\angle YUX\end{align*}), \begin{align*}\angle YUZ\end{align*} (or \begin{align*}\angle ZUY\end{align*}), and \begin{align*}\angle XUZ\end{align*} (or \begin{align*}\angle ZUX\end{align*}).
Protractor Postulate
We measure a line segment’s length with a ruler. Angles are measured with something called a protractor. A protractor is a measuring device that measures how “open” an angle is. Angles are measured in degrees, and labeled with a \begin{align*}^\circ\end{align*} symbol.
There are two sets of measurements, one starting on the left and the other on the right side of the protractor. Both go around from \begin{align*}0^\circ\end{align*} to \begin{align*}180^\circ\end{align*}. When measuring angles, always line up one side with \begin{align*}0^\circ\end{align*}, and see where the other side hits the protractor. The vertex lines up in the middle of the bottom line.
Example 2: Measure the three angles from Example 1, using a protractor.
Solution: Just like in Example 1, it might be easier to measure these three angles if we separate them.
With measurement, we put an \begin{align*}m\end{align*} in front of the \begin{align*}\angle\end{align*} sign to indicate measure. So, \begin{align*}m\angle XUY = 84^\circ, \ m\angle YUZ = 42^\circ\end{align*} and \begin{align*}m\angle XUZ = 126^\circ\end{align*}.
Just like the Ruler Postulate for line segments, there is a Protractor Postulate for angles.
Protractor Postulate: For every angle there is a number between \begin{align*}0^\circ\end{align*} and \begin{align*}180^\circ\end{align*} that is the measure of the angle. The angle's measure is the difference of the degrees where the sides of the angle intersect the protractor. For now, angles are always positive.
In other words, you do not have to start measuring an angle at \begin{align*}0^\circ\end{align*}, as long as you subtract one measurement from the other.
Example 3: What is the measure of the angle shown below?
Solution: This angle is lined up with \begin{align*}0^\circ\end{align*}, so where the second side intersects the protractor is the angle measure, which is \begin{align*}50^\circ\end{align*}.
Example 4: What is the measure of the angle shown below?
Solution: This angle is not lined up with \begin{align*}0^\circ\end{align*}, so use subtraction to find its measure. It does not matter which scale you use.
Inner scale: \begin{align*}140^\circ - 25^\circ = 125^\circ\end{align*}
Outer scale: \begin{align*}165^\circ - 40^\circ = 125^\circ\end{align*}
Example 5: Use a protractor to measure \begin{align*}\angle RST\end{align*} below.
Solution: Lining up one side with \begin{align*}0^\circ\end{align*} on the protractor, the other side hits \begin{align*}100^\circ\end{align*}.
Classifying Angles
Angles can be grouped into four different categories.
Straight Angle: An angle that measures exactly \begin{align*}180^\circ\end{align*}.
Right Angle: An angle that measures exactly \begin{align*}90^\circ\end{align*}.
This half-square marks right, or \begin{align*}90^\circ\end{align*}, angles.
Acute Angles: Angles that measure between \begin{align*}0^\circ\end{align*} and \begin{align*}90^\circ\end{align*}.
Obtuse Angles: Angles that measure between \begin{align*}90^\circ\end{align*} and \begin{align*}180^\circ\end{align*}.
Perpendicular: When two lines intersect to form four right angles.
Even though all four angles are \begin{align*}90^\circ\end{align*}, only one needs to be marked with the half-square.
The symbol for perpendicular is \begin{align*}\perp\end{align*}.
Label It | Say It |
---|---|
\begin{align*}l \perp m\end{align*} | Line \begin{align*}l\end{align*} is perpendicular to line \begin{align*}m\end{align*}. |
\begin{align*}\overleftrightarrow{AC} \perp \overleftrightarrow{DE}\end{align*} | Line \begin{align*}AC\end{align*} is perpendicular to line \begin{align*}DE\end{align*}. |
Example 6: Name the angle and determine what type of angle it is.
Solution: The vertex is \begin{align*}U\end{align*}. So, the angle can be \begin{align*}\angle TUV\end{align*} or \begin{align*}\angle VUT\end{align*}. To determine what type of angle it is, compare it to a right angle.
Because it opens wider than a right angle, and less than a straight angle it is obtuse.
Example 7: What type of angle is \begin{align*}84^\circ\end{align*}? What about \begin{align*}165^\circ\end{align*}?
Solution: \begin{align*}84^\circ\end{align*} is less than \begin{align*}90^\circ\end{align*}, so it is acute. \begin{align*}165^\circ\end{align*} is greater than \begin{align*}90^\circ\end{align*}, but less than \begin{align*}180^\circ\end{align*}, so it is obtuse.
Drawing an Angle
Investigation 1-2: Drawing a \begin{align*}50^\circ\end{align*} Angle with a Protractor
1. Start by drawing a horizontal line across the page, 2 in long.
2. Place an endpoint at the left side of your line.
3. Place the protractor on this point, such that the bottom line of the protractor is on the line and the endpoint is at the center. Mark \begin{align*}50^\circ\end{align*} on the appropriate scale.
4. Remove the protractor and connect the vertex and the \begin{align*}50^\circ\end{align*} mark.
This process can be used to draw any angle between \begin{align*}0^\circ\end{align*} and \begin{align*}180^\circ\end{align*}. See http://www.mathsisfun.com/geometry/protractor-using.html for an animation of this investigation.
Example 8: Draw a \begin{align*}135^\circ\end{align*} angle.
Solution: Following the steps from above, your angle should look like this:
Now that we know how to draw an angle, we can also copy that angle with a compass and a ruler. Anytime we use a compass and ruler to draw geometric figures, it is called a construction.
Compass: A tool used to draw circles and arcs.
Investigation 1-3: Copying an Angle with a Compass and Ruler
1. We are going to copy the \begin{align*}50^\circ\end{align*} angle from Investigation 1-2. First, draw a straight line, 2 inches long, and place an endpoint at one end.
2. With the point (non-pencil side) of the compass on the vertex, draw an arc that passes through both sides of the angle. Repeat this arc with the line we drew in #1.
3. Move the point of the compass to the horizontal side of the angle we are copying. Place the point where the arc intersects this side. Open (or close) the “mouth” of the compass so that you can draw an arc that intersects the other side and the arc drawn in #2. Repeat this on the line we drew in #1.
4. Draw a line from the new vertex to the arc intersections.
To watch an animation of this construction, see http://www.mathsisfun.com/geometry/construct-anglesame.html
Marking Angles and Segments in a Diagram
With all these segments and angles, we need to have different ways to label equal angles and segments.
Angle Markings
Segment Markings
Example 9: Write all equal angle and segment statements.
Solution: \begin{align*}\overline{AD} \perp \overleftrightarrow{FC}\end{align*}
\begin{align*}m\angle ADB &= m \angle BDC = m \angle FDE = 45^\circ\\ AD &= DE\\ FD &= DB = DC\\ m \angle ADF &= m \angle ADC = 90^\circ\end{align*}
Angle Addition Postulate
Like the Segment Addition Postulate, there is an Angle Addition Postulate.
Angle Addition Postulate: If \begin{align*}B\end{align*} is on the interior of \begin{align*}\angle ADC\end{align*}, then \begin{align*}m \angle ADC = m \angle ADB + m \angle BDC\end{align*}.
Example 10: What is \begin{align*}m \angle QRT\end{align*} in the diagram below?
Solution: Using the Angle Addition Postulate, \begin{align*}m \angle QRT = 15^\circ + 30^\circ = 45^\circ\end{align*}.
Example 11: What is \begin{align*}m \angle LMN\end{align*} if \begin{align*}m \angle LMO = 85^\circ\end{align*} and \begin{align*}m \angle NMO = 53^\circ\end{align*}?
Solution: \begin{align*}m\angle LMO = m\angle NMO + m\angle LMN\end{align*}, so \begin{align*}85^\circ = 53^\circ+m\angle LMN\end{align*}.
\begin{align*}m \angle LMN = 32^\circ.\end{align*}
Example 12: Algebra Connection If \begin{align*}m\angle ABD = 100^\circ\end{align*}, find \begin{align*}x\end{align*}.
Solution: \begin{align*}m\angle ABD = m\angle ABC + m\angle CBD\end{align*}. Write an equation.
\begin{align*}100^\circ & = (4x+2)^\circ + (3x-7)^\circ\\ 100^\circ & = 7x^\circ-5^\circ\\ 105^\circ & = 7x^\circ\\ 15^\circ & = x\end{align*}
Know What? Revisited Using a protractor, the measurement marked in the red triangle is \begin{align*}90^\circ\end{align*}, the measurement in the blue triangle is \begin{align*}45^\circ\end{align*} and the measurement in the orange square is \begin{align*}90^\circ\end{align*}.
Review Questions
- Questions 1-10 use the definitions, postulates and theorems from this section.
- Questions 11-16 are similar to Investigation 1-2 and Examples 7 and 8.
- Questions 17 and 18 are similar to Investigation 1-3.
- Questions 19-22 are similar to Examples 2-5.
- Question 23 is similar to Example 9.
- Questions 24-28 are similar to Examples 10 and 11.
- Questions 29 and 30 are similar to Example 12.
For questions 1-10, determine if the statement is true or false.
- Two angles always add up to be greater than \begin{align*}90^\circ\end{align*}.
- \begin{align*}180^\circ\end{align*} is an obtuse angle.
- \begin{align*}180^\circ\end{align*} is a straight angle.
- Two perpendicular lines intersect to form four right angles.
- A construction uses a protractor and a ruler.
- For an angle \begin{align*}\angle ABC, C\end{align*} is the vertex.
- For an angle \begin{align*}\angle ABC, \overline{AB}\end{align*} and \begin{align*}\overline{BC}\end{align*} are the sides.
- The \begin{align*}m\end{align*} in front of \begin{align*}m \angle ABC\end{align*} means measure.
- Angles are always measured in degrees.
- The Angle Addition Postulate says that an angle is equal to the sum of the smaller angles around it.
For 11-16, draw the angle with the given degree, using a protractor and a ruler. Also, state what type of angle it is.
- \begin{align*}55^\circ\end{align*}
- \begin{align*}92^\circ\end{align*}
- \begin{align*}178^\circ\end{align*}
- \begin{align*}5^\circ\end{align*}
- \begin{align*}120^\circ\end{align*}
- \begin{align*}73^\circ\end{align*}
- Construction Copy the angle you made from #12, using a compass and a ruler.
- Construction Copy the angle you made from #16, using a compass and a ruler.
For 19-22, use a protractor to determine the measure of each angle.
- Interpret the picture to the right. Write down all equal angles, segments and if any lines are perpendicular.
In Exercises 24-29, use the following information: \begin{align*}Q\end{align*} is in the interior of \begin{align*}\angle ROS\end{align*}. \begin{align*}S\end{align*} is in the interior of \begin{align*}\angle QOP\end{align*}. \begin{align*}P\end{align*} is in the interior of \begin{align*}\angle SOT\end{align*}. \begin{align*}S\end{align*} is in the interior of \begin{align*}\angle ROT\end{align*} and \begin{align*}m\angle ROT = 160^\circ, \ m\angle SOT = 100^\circ\end{align*}, and \begin{align*}m\angle ROQ = m\angle QOS = m\angle POT\end{align*}.
- Make a sketch.
- Find \begin{align*}m\angle QOP\end{align*}
- Find \begin{align*}m\angle QOT\end{align*}
- Find \begin{align*}m\angle ROQ\end{align*}
- Find \begin{align*}m\angle SOP\end{align*}
Algebra Connection Solve for \begin{align*}x\end{align*}.
- \begin{align*}m\angle ADC = 56^\circ\end{align*}
- \begin{align*}m\angle ADC = 130^\circ\end{align*}
Review Queue Answers
1. \begin{align*}\overrightarrow{AB}\end{align*}, a ray
2. \begin{align*}XY = 3, YZ = 38\end{align*}
\begin{align*}{\;} \ a-6+3a+11= 41\!\\ {\;}\qquad \qquad 4a+5 =41\!\\ {\;}\qquad \qquad \quad \ \ 4a = 36\!\\ {\;}\qquad \qquad \quad \ \ \ a = 9\end{align*}
3. Use the Segment Addition Postulate, \begin{align*}AC = 13\end{align*}.