Lesson 3

Nonadjacent Angles

Let’s look at angles that are not right next to one another.

3.1: Finding Related Statements

Given \(a\) and \(b\) are numbers, and \(a+b=180\), which statements also must be true?

\(a=180-b\)

\(a-180=b\)

\(360=2a+2b\)

\(a=90\) and \(b=90\)

3.2: Polygon Angles

Use any useful tools in the geometry toolkit to identify any pairs of angles in these figures that are complementary or supplementary.

A quadrilateral and a triangle.  Please ask for futher assistance.

 

3.3: Vertical Angles

Use a straightedge to draw two intersecting lines. Use a protractor to measure all four angles whose vertex is located at the intersection.

 
Compare your drawing and measurements to the people in your group. Make a conjecture about the relationships between angle measures at an intersection.

3.4: Row Game: Angles

Find the measure of the angles in one column. Your partner will work on the other column. Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to find the error and correct it.

column A

column B

\(P\) is on line \(m\). Find the value of \(a\).

A ray from point P, on line m, creates 2 angles, 134 degrees and a, degrees.

Find the value of \(b\).

A ray from the vertex of a right angle creates two angles, 44 degrees, b degrees.

Find the value of \(a\).

Two lines are perpendicular. A ray from the point where they meet creates 2 angles, 51 degrees, a, degrees.

In right triangle \(LMN\), angles \(L\) and \(M\) are complementary. Find the measure of angle \(L\).

A right triangle L M N. Angle N is 90 degrees, angle M is 51 degrees.

 

column A

column B

Angle \(C\) and angle \(E\) are supplementary. Find the measure of angle \(E\).

Quadrilateral C D E F. Angle C, 129 degrees.

\(X\) is on line \(WY\). Find the value of \(b\).

Point X is on line W Y. Rays V X and U X form 3 angles, b degrees, 95 degrees, 34 degrees.

Find the value of \(c\).

Three lines meet, forming 6 angles. The measures are, clockwise, c degrees, 42 degrees, 90 degress, d degrees, b degrees, 90 degrees.

\(B\) is on line \(FW\). Find the measure of angle \(CBW\).

Three lines meet, forming 6 angles.  The measures are 67 degrees, 65 degrees, blank, blank, blank, blank.

Two angles are complementary. One angle measures 37 degrees. Find the measure of the other angle.

Two angles are supplementary. One angle measures 127 degrees. Find the measure of the other angle.

 

Summary

When two lines cross, they form two pairs of vertical angles. Vertical angles are across the intersection point from each other.

Two lines cross, with the 4 angles formed marked.

Vertical angles always have equal measure. We can see this because they are always supplementary with the same angle. For example:

Two images. Both images intersecting lines, an obtuse angle, 150 degrees.  One image, the angle adjacent counter-clockwise is 30 degrees, the other image, the angle adjacent clockwise is 30 degrees.

This is always true!

Two images. Both images intersecting lines, an obtuse angle, b.  One image, the angle adjacent counter-clockwise is a, degrees, the other image, the angle adjacent clockwise is c degrees.

\(a+b = 180\) so \(a = 180-b\).

\(c+b = 180\) so \(c = 180-b\).

That means \(a = c\).

Glossary Entries

  • adjacent angles

    Adjacent angles share a side and a vertex.

    In this diagram, angle \(ABC\) is adjacent to angle \(DBC\).

    Three segments all joined at endpoint B. Point A is to the left of B and segment A B is drawn. Point C is above B and segment C B is drawn. Point D is to the right of B and segment B D is drawn.
  • complementary

    Complementary angles have measures that add up to 90 degrees.

    For example, a \(15^\circ\) angle and a \(75^\circ\) angle are complementary.

    complementary angles of 15 and 75 degrees
    Two angles, one is 75 degrees and one is 15 degrees
  • right angle

    A right angle is half of a straight angle. It measures 90 degrees.

    a right angle
  • straight angle

    A straight angle is an angle that forms a straight line. It measures 180 degrees.

    a 180 degree angle
  • supplementary

    Supplementary angles have measures that add up to 180 degrees.

    For example, a \(15^\circ\) angle and a \(165^\circ\) angle are supplementary.

    supplementary angles of 15 and 165 degrees
    supplementary angles of 15 and 165 degrees
  • vertical angles

    Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.

    For example, angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^\circ\), then angle \(DEB\) must also measure \(120^\circ\).

    Angles \(AED\) and \(BEC\) are another pair of vertical angles.

    a pair of intersecting lines that create vertical angles