<meta http-equiv="refresh" content="1; url=/nojavascript/"> Angle Pairs | CK-12 Foundation
Dismiss
Skip Navigation
You are reading an older version of this FlexBook® textbook: CK-12 Geometry - Basic Go to the latest version.

1.5: Angle Pairs

Created by: CK-12

Learning Objectives

  • Recognize complementary angles supplementary angles, linear pairs, and vertical angles.
  • Apply the Linear Pair Postulate and the Vertical Angles Theorem.

Review Queue

  1. Find x.
  2. Find y.
  3. Find z.

Know What? A compass (as seen to the right) is used to determine the direction a person is traveling. The angles between each direction are very important because they enable someone to be more specific with their direction. A direction of 45^\circ \ NW, would be straight out along that northwest line.

What headings have the same angle measure? What is the angle measure between each compass line?

Complementary Angles

Complementary: Two angles that add up to 90^\circ.

Complementary angles do not have to be:

  • congruent
  • next to each other

Example 1: The two angles below are complementary. m\angle GHI = x. What is x?

Solution: Because the two angles are complementary, they add up to 90^\circ. Make an equation.

x + 34^\circ & = 90^\circ\\x & = 56^\circ

Example 2: The two angles below are complementary. Find the measure of each angle.

Solution: The two angles add up to 90^\circ. Make an equation.

8r + 9^\circ + 7r + 6^\circ & = 90^\circ\\15r + 15^\circ & = 90^\circ\\15r & = 75^\circ\\r & = 5^\circ

However, you need to find each angle. Plug r back into each expression.

m \angle GHI & = 8(5^\circ) + 9^\circ = 49^\circ\\m\angle JKL & = 7(5^\circ) + 6^\circ = 41^\circ

Supplementary Angles

Supplementary: Two angles that add up to 180^\circ.

Supplementary angles do not have to be:

  • congruent
  • next to each other

Example 3: The two angles below are supplementary. If m\angle MNO = 78^\circ what is m\angle PQR?

Solution: Set up an equation. However, instead of equaling 90^\circ, now it is 180^\circ.

78^\circ + m\angle PQR & = 180^\circ\\m\angle PQR & = 102^\circ

Example 4: What is the measure of two congruent, supplementary angles?

Solution: Supplementary angles add up to 180^\circ. Congruent angles have the same measure. So, 180^\circ \div 2 = 90^\circ, which means two congruent, supplementary angles are right angles, or 90^\circ.

Linear Pairs

Adjacent Angles: Two angles that have the same vertex, share a side, and do not overlap.

\angle PSQ and \angle QSR are adjacent.

\angle PQR and \angle PQS are NOT adjacent because they overlap.

Linear Pair: Two angles that are adjacent and the non-common sides form a straight line.

\angle PSQ and \angle QSR are a linear pair.

Linear Pair Postulate: If two angles are a linear pair, then they are supplementary.

Example 5: Algebra Connection What is the measure of each angle?

Solution: These two angles are a linear pair, so they add up to 180^\circ.

(7q-46)^\circ + (3q+6)^\circ &= 180^\circ\\10q - 40^\circ &= 180^\circ\\10q & = 220^\circ\\q & = 22^\circ

Plug in q to get the measure of each angle. m\angle ABD = 7(22^\circ) - 46^\circ = 108^\circ \ m\angle DBC = 180^\circ - 108^\circ = 72^\circ

Example 6: Are \angle CDA and \angle DAB a linear pair? Are they supplementary?

Solution: The two angles are not a linear pair because they do not have the same vertex. They are supplementary, 120^\circ + 60^\circ = 180^\circ.

Vertical Angles

Vertical Angles: Two non-adjacent angles formed by intersecting lines.

\angle 1 and \angle 3 are vertical angles

\angle 2 and \angle 4 are vertical angles

These angles are labeled with numbers. You can tell that these are labels because they do not have a degree symbol.

Investigation 1-6: Vertical Angle Relationships

  1. Draw two intersecting lines on your paper. Label the four angles created \angle 1, \ \angle 2, \ \angle 3, and \angle 4, just like the picture above.
  2. Use your protractor to find m\angle 1.
  3. What is the angle relationship between \angle 1 and \angle 2 called? Find m\angle 2.
  4. What is the angle relationship between \angle 1 and \angle 4 called? Find m\angle 4.
  5. What is the angle relationship between \angle 2 and \angle 3 called? Find m\angle 3.
  6. Are any angles congruent? If so, write them down.

From this investigation, you should find that \angle 1 \cong \angle 3 and \angle 2 \cong \angle 4.

Vertical Angles Theorem: If two angles are vertical angles, then they are congruent.

We can prove the Vertical Angles Theorem using the same process we used in the investigation. We will not use any specific values for the angles.

From the picture above:

\angle 1 \ \text{and} \ \angle 2 \ \text{are a linear pair} \ \rightarrow m\angle 1 + m\angle 2 & = 180^\circ \qquad \text{Equation} \ 1\\\angle 2 \ \text{and} \ \angle 3 \ \text{are a linear pair} \ \rightarrow m\angle 2 + m\angle 3 & = 180^\circ \qquad \text{Equation} \ 2\\\angle 3 \ \text{and} \ \angle 4 \ \text{are a linear pair} \ \rightarrow m\angle 3 + m\angle 4 & = 180^\circ \qquad \text{Equation} \ 3

All of the equations = 180^\circ, so Equation 1 = Equation 2 and Equation 2 = Equation 3.

m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \qquad \text{and} \qquad m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4

Cancel out the like terms

m\angle 1 = m\angle 3 \qquad \text{and} \qquad m\angle 2 = m\angle 4

Recall that anytime the measures of two angles are equal, the angles are also congruent. So, \angle 1 \cong \angle 3 and \angle 2 \cong \angle 4 too.

Example 7: Find m\angle 1 and m\angle 2.

Solution: \angle 1 is vertical angles with 18^\circ, so m\angle 1 = 18^\circ.

\angle 2 is a linear pair with \angle 1 or 18^\circ, so 18^\circ + m\angle 2 = 180^\circ.

m\angle 2 = 180^\circ - 18^\circ = 162^\circ.

Know What? Revisited The compass has several vertical angles and all of the smaller angles are 22.5^\circ, 180^\circ \div 8. Directions that are opposite each other have the same angle measure, but of course, a different direction. All of the green directions have the same angle measure, 22.5^\circ, and the purple have the same angle measure, 45^\circ. N, \ S, \ E and W all have different measures, even though they are all 90^\circ apart.

Review Questions

  • Questions 1 and 2 are similar to Examples 1, 2, and 3.
  • Questions 3-8 are similar to Examples 3, 4, 6 and 7.
  • Questions 9-16 use the definitions, postulates and theorems from this section.
  • Questions 17-25 are similar to Example 5.
  1. Find the measure of an angle that is complementary to \angle ABC if m\angle ABC is
    1. 45^\circ
    2. 82^\circ
    3. 19^\circ
    4. z^\circ
  2. Find the measure of an angle that is supplementary to \angle ABC if m\angle ABC is
    1. 45^\circ
    2. 118^\circ
    3. 32^\circ
    4. x^\circ

Use the diagram below for exercises 3-7. Note that \overline{NK} \perp \overleftrightarrow{IL}.

  1. Name one pair of vertical angles.
  2. Name one linear pair of angles.
  3. Name two complementary angles.
  4. Name two supplementary angles.
  1. What is:
    1. m\angle INL
    2. m\angle LNK
  2. If m\angle INJ = 63^\circ, find:
    1. m\angle JNL
    2. m\angle KNJ
    3. m\angle MNL
    4. m\angle MNI

For 9-16, determine if the statement is true or false.

  1. Vertical angles are congruent.
  2. Linear pairs are congruent.
  3. Complementary angles add up to 180^\circ.
  4. Supplementary angles add up to 180^\circ
  5. Adjacent angles share a vertex.
  6. Adjacent angles overlap.
  7. Complementary angles are always 45^\circ.
  8. Vertical angles have the same vertex.

For 17-25, find the value of x or y.

  1. Find x.
  2. Find y.

Review Queue Answers

  1. x+26 = 3x-8\!\\{\;} \quad \ 34 = 2x\!\\{\;} \quad \ 17 = x
  2. (7y+6)^\circ = 90^\circ\!\\{\;} \qquad \ 7y = 84^\circ\!\\{\;} \qquad \ \ y = 12^\circ
  3. z+ 15 = 5z + 9\!\\{\;} \quad \ \ 6 = 4z\!\\{\;} \quad 1.5 = z

Image Attributions

Files can only be attached to the latest version of None

Reviews

Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
CK.MAT.ENG.SE.1.Geometry-Basic.1.5

Original text