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1.3: Angles and Measurement

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Learning Objectives

  • Define and classify angles.
  • Apply the Protractor Postulate and the Angle Addition Postulate.

Review Queue

Answer the following questions.

  1. Label the following geometric figure. What is it called?
  2. Find a, XY and YZ.
  3. Find x, CD and DE.
  4. B is between A and C on \overline{AC}. If AB = 4 and BC = 9, what is AC? What postulate do you use to solve this problem?

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. Also, determine which other angles are equal to these measurements. Use appropriate angle markings. Do not measure any arcs.

Two Rays = One Angle

In #1 above, the figure was a ray. It is labeled \overrightarrow{AB}, with the arrow over the point that is NOT the endpoint. When two rays have the same endpoint, an angle is created.

Here, \overrightarrow{BA} and \overrightarrow{BC} meet to form an angle. An angle is labeled with an “\angle” symbol in front of the three letters used to label it. This angle can be labeled \angle ABC or \angle CBA. Always put the vertex in the middle of the three points. It doesn’t matter which side point is written first.

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.

Example 1: How many angles are in the picture below? Label each one two different ways.

Solution: There are three angles with vertex U. It might be easier to see them all if we separate them out.

So, the three angles can be labeled, \angle XUY or  \angle YUX, \ \angle YUZ or  \angle ZUY, and \angle XUZ or  \angle ZUX.

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 ^\circ symbol.

Notice that there are two sets of measurements, one opening clockwise and one opening counter-clockwise, from 0^\circ to 180^\circ. When measuring angles, always line up one side with 0^\circ, and see where the other side hits the protractor. The vertex lines up in the middle of the bottom line, where all the degree lines meet.

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 you separate them. With measurement, we put an m in front of the \angle sign to indicate measure. So, m \angle XUY = 84^\circ, \ m \angle YUZ = 42^\circ and m \angle XUZ = 126^\circ.

In the last lesson, we introduced the Ruler Postulate. Here we introduce the Protractor Postulate.

Protractor Postulate: For every angle there is a number between 0^\circ and 180^\circ that is the measure of the angle in degrees. The angle's measure is then the absolute value of the difference of the numbers shown on the protractor where the sides of the angle intersect the protractor.

In other words, you do not have to start measuring an angle at 0^\circ, 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 not lined up with 0^\circ, so use subtraction to find its measure. It does not matter which scale you use.

Using the inner scale, |140 - 25| = 125^\circ

Using the outer scale, |165 - 40| = 125^\circ

Example 4: Use a protractor to measure \angle RST below.

Solution: The easiest way to measure any angle is to line one side up with 0^\circ. This angle measures 100^\circ.

Classifying Angles

By looking at the protractor we measure angles from 0^\circ to 180^\circ. Angles can be classified, or grouped, into four different categories.

Straight Angle: When an angle measures 180^\circ. The angle measure of a straight line. The rays that form this angle are called opposite rays.

Right Angle: When an angle measures 90^\circ.

Notice the half-square, marking the angle. This marking is always used to mark right, or 90^\circ, angles.

Acute Angles: Angles that measure between 0^\circ and 90^\circ.

Obtuse Angles: Angles that measure between 90^\circ and 180^\circ.

It is important to note that 90^\circ is NOT an acute angle and 180^\circ is NOT an obtuse angle.

Additionally, any two lines or line segments can intersect to form four angles. If the two lines intersect to form right angles, we say the lines are perpendicular.

Perpendicular: When two lines intersect to form four right angles.

Even though all four angles are 90^\circ, only one needs to be marked. It can be assumed thatall four are 90^\circ.

The symbol for perpendicular is \bot, so these two lines would be labeled l \bot m or \overleftrightarrow{A C} \bot \overleftrightarrow{D E}.

There are several other ways to label these two intersecting lines. This picture shows two perpendicular lines, four right angles, four 90^\circ angles, and even two straight angles, \angle ABC and \angle DBE.

Example 5: Name the angle and determine what type of angle it is.

Solution: The vertex is U. So, the angle can be \angle TUV or \angle VUT. 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 6: What type of angle is 84^\circ? What about 165^\circ?

Solution: 84^\circ is less than 90^\circ, so it is acute. 165^\circ is greater than 90^\circ, but less than 180^\circ, so it is obtuse.

Drawing an Angle

Investigation 1-1: Drawing a 50^\circ Angle with a Protractor

  1. Start by drawing a horizontal line across the page, about 2 in long.
  2. Place an endpoint at the left side of your line.
  3. Place the protractor on this point. Make sure to put the center point on the bottom line of the protractor on the vertex. Mark 50^\circ on the appropriate scale.
  4. Remove the protractor and connect the vertex and the 50^\circ mark.

This process can be used to draw any angle between 0^\circ and 180^\circ. See http://www.mathsisfun.com/geometry/protractor-using.html for an animation of this investigation.

Example 7: Draw a 135^\circ 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 straightedge, usually a ruler. Anytime we use a compass and ruler to draw different geometric figures, it called a construction.

Compass: A tool used to draw circles and arcs.

Investigation 1-2: Copying an Angle with a Compass and Straightedge

  1. We are going to copy the angle created in the previous investigation, a 50^\circ angle. First, draw a straight line, about 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 you can draw an arc that intersects the other side of 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 8: Interpret the picture below. Write all equal angle and segment statements.

Solution:

& \overline{AD} \bot \overleftrightarrow{FC}\\& m \angle ADB = m \angle BDC = m \angle FDE = 45^\circ\\& AD = DE\\& FD = DB = DC\\& m \angle ADF = m \angle ADC = 90^\circ

Angle Addition Postulate

Much like the Segment Addition Postulate, there is an Angle Addition Postulate.

Angle Addition Postulate: If B is on the interior of \angle ADC, then m \angle ADC = m \angle ADB + m \angle BDC. See the picture below.

Example 9: What is m \angle QRT in the diagram below?

Solution: Using the Angle Addition Postulate, m \angle QRT = 15^\circ + 30^\circ = 45^\circ.

Example 10: What is m \angle LMN if m \angle LMO = 85^\circ and m \angle NMO = 53^\circ?

Solution: From the Angle Addition Postulate,  m \angle LMO = m \angle NMO + m \angle LMN. Substituting in what we know, 85^\circ = 53^\circ + m \angle LMN, so 85^\circ - 53^\circ = m \angle LMN or m \angle LMN = 32^\circ.

Example 11: Algebra Connection If m \angle ABD = 100^\circ, find x and m \angle ABC and m \angle CBD?

Solution: From the Angle Addition Postulate, m \angle ABD = m \angle ABC + m \angle CBD. Substitute in what you know and solve the equation.

100^\circ & = (4x + 2)^\circ + (3x - 7)^\circ\\100^\circ & = 7x - 5^\circ\\105^\circ & = 7x\\15^\circ & = x

So, m \angle ABC = 4(15^\circ) + 2^\circ = 62^\circ and m \angle CBD = 3(15^\circ) - 7^\circ = 38^\circ.

Know What? Revisited Using a protractor, the measurement marked in the red triangle is 90^\circ, the measurement in the blue triangle is 45^\circ and the measurement in the orange square is 90^\circ.

All of the equal angles are marked in the picture to the right. All of the acute angles in the triangles are equal and all the other angles are right, or 90^\circ.

Review Questions

For questions 1-10, determine if the statement is true or false. If you answered FALSE for any question, state why.

  1. Two angles always add up to be greater than 90^\circ.
  2. 180^\circ is an obtuse angle.
  3. 180^\circ is a straight angle.
  4. Two perpendicular lines intersect to form four right angles.
  5. A construction uses a protractor and a ruler.
  6. For an angle \angle ABC, C is the vertex.
  7. For an angle \angle ABC, \overline{AB} and \overline{BC} are the sides.
  8. The m in front of m \angle ABC means measure.
  9. Angles are always measured in degrees.
  10. 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.

  1. 55^\circ
  2. 92^\circ
  3. 178^\circ
  4. 5^\circ
  5. 120^\circ
  6. 73^\circ
  7. Construction Copy the angle you made from #12, using a compass and a straightedge.
  8. Construction Copy the angle you made from #16, using a compass and a straightedge.

For 19-22, use a protractor to determine the measure of each angle.

  1. Interpret the picture to the right. Write down all equal angles, segments and if any lines are perpendicular.
  2. Draw a picture with the following requirements.

&AB = BC = BD && m \angle ABD = 90^\circ\\&m \angle ABC = m \angle CBD && A, B, C \ \text{and} \ D \ \text{are coplanar}

In 25 and 26, plot and sketch \angle ABC. Classify the angle. Write the coordinates of a point that lies in the interior of the angle.

  1. A(5, -3)\!\\B(-3, -1)\!\\C(2, 2)
  2. A(-3, 0)\!\\B(1, 3)\!\\C(5, 0)

In Exercises 27-31, use the following information: Q is in the interior of \angle ROS. S is in the interior of \angle QOP. P is in the interior of \angle SOT. S is in the interior of \angle ROT and m \angle ROT = 160^\circ, \ m \angle SOT = 100^\circ, and m \angle ROQ = m \angle QOS = m \angle POT.

  1. Make a sketch.
  2. Find m \angle QOP
  3. Find m \angle QOT
  4. Find m \angle ROQ
  5. Find m \angle SOP

Algebra Connection Solve for x.

  1. m \angle ADC = 56^\circ
  2. m \angle ADC = 130^\circ
  3. m \angle ADC = (16x - 55)^\circ
  4. m \angle ADC = ( 9x - 80)^\circ
  5. Writing Write a paragraph about why the degree measure of a straight line is 180, the degree measure of a right angle is 90, etc. In other words, answer the question, “Why is the straight line divided into exactly 180 degrees and not some other number of degrees?”

Review Queue Answers

1. \overrightarrow{AB}, a ray

2. XY = 3, \ YZ = 38\!\\a - 6 + 3a + 11 = 41\!\\{\;} \qquad \quad \ 4a + 5 = 41\!\\{\;} \qquad \qquad \ \ \ 4a = 36\!\\{\;} \qquad  \qquad \ \ \ \ a = 9

3. CD = 51, \ DE = 10\!\\8x + 3 + 3x - 8 = 4x + 37\!\\{\;} \qquad \quad \ 11x - 5 = 4x + 37\!\\{\;} \qquad \qquad \ \ \ \ 7x = 42\!\\{\;} \qquad \qquad \quad \ \ x = 6

4. Use the Segment Addition Postulate, AC = 13.

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