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1. Foundations of Geometry
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United States of AmericaFL
Grade 912

1.04 Angle relationships

Lesson
Practice
Lesson

Concept summary

Certain pairs of angles can have a special relationship:

  • Complementary angles have a sum of 90\degree
  • Supplementary angles have a sum of 180\degree
Linear pair

Adjacent angles that form a straight line.

A straight horizontal line with a ray extending from a point on the line. The ray creates two angles.

The linear pair postulate states that if two angles form a linear pair, then they are supplementary.

Vertical angles

The opposite angles formed when two lines intersect.

Two lines intersecting and creating two pairs of angles. The pair of angles that are across from each other and have the larger measure are marked with one congruent marking, the pair of angles with the smaller measure that are across from each other are marked with two congruent markings.

The vertical angles theorem states that vertical angles are congruent.

Angle bisector

A line, segment or ray that divides an angle into two congruent angles.

An angle with a ray inside that cuts it into two smaller congruent angles.
Perpendicular lines

Two lines that intersect at right angles.

Two intersecting lines with a right angle marking.
Perpendicular

Intersecting to form 90 \degree angles, denoted by \perp.

Two intersecting lines with a right angle marking.
Perpendicular bisector

A line, segment, or ray that is perpendicular to a segment at its midpoint.

A vertical line passing through a horizontal line segment. The line splits the segment into two smaller congruent segments. The angle where the line and the segment intersect has a right angle marking.

To construct the perpendicular bisector of a segment, we will use the same set of steps as for bisecting a segment:

  1. Identify the segment we want to bisect.
  2. Open the compass width to just past half the segment's length and draw an arc from one endpoint that extends to both sides of the segment.
  3. Without changing the compass width, draw another arc from the other endpoint that intersects the original arc on both sides of the segment.
  4. Label the intersection of the arcs with points.
  5. Connect the points.
A diagram showing the 5 steps of constructing the bisector of a segment. Speak to your teacher for more information.

To construct the bisector of an angle, we will:

  1. Identify the angle we want to bisect.
  2. With the compass point on the vertex of the angle, use the compass to draw an arc that intersects both legs.
  3. Label the intersections with points.
  4. With the compass point on one of the points from Step 3, draw an arc that passes halfway through the interior of the angle.
  5. With the compass point on the other point from Step 3, draw an arc that passes halfway through the interior of the angle and intersects the first arc.
  6. Label the intersection of the arcs drawn in parts 4 and 5 with a point.
  7. Draw a line that connects the vertex of the angle and the point added in Step 6.
A diagram showing the 7 steps of constructing the bisector of an angle. Speak to your teacher for more information.

Worked examples

Example 1

The angles in the diagram are complementary. Find the value of x.

A right angle with a point in the interior of the angle. A segment is drawn from the vertex of the angle to the point, forming two adjacent angles. The measure of the two angles are x degrees, and 39 degrees.

Approach

Complementary angles have a sum of 90\degree. Use this to write an equation that includes the two angles in the diagram knowing that the angles are complementary. Then we want to solve for x.

Solution

\displaystyle x+39\displaystyle =\displaystyle 90Definition of complementary
\displaystyle x\displaystyle =\displaystyle 51Subtract

x=51 \degree

Reflection

If the angles were supplementary, then the sum of the angles would equal 180 \degree insead of 90 \degree.

Example 2

Use the diagram to identify an example of each angle pair.

Lines A D and E C intersecting at point F. Ray F B is in the interior of angle A F C. Angle B F C has a measure of 28 degrees, angle A F E has a measure of 62 degrees, and angle E F D has a measure of 118 degrees.
a

Vertical angles

Approach

Vertical angles are formed by intersecting lines. There is only one pair of intersecting lines in the diagram, \overleftrightarrow{AD} and \overleftrightarrow{CE}. Identify a pair of opposite angles formed by this intersection.

Solution

\angle{AFC} and \angle{EFD} or \angle{AFE} and \angle{CFD}

b

Linear pair

Approach

Linear pairs are adjacent angles that form a line. First, we need to identify a line such as \overleftrightarrow{AD} or \overleftrightarrow{CE} and see if we can identify adjacent angles that form this line.

Solution

\angle{AFE} and \angle{EFD} form a linear pair.

Reflection

There are multiple linear pairs in the diagram.

Outcomes

MA.912.GR.1.1

Prove relationships and theorems about lines and angles. Solve mathematical and real-world problems involving postulates, relationships and theorems of lines and angles.

MA.912.GR.5.2

Construct the bisector of a segment or an angle, including the perpendicular bisector of a line segment.

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