Topics
1. Foundations of Geometry
topic badge
United States of AmericaFL
Grade 912

1.03 Measuring angles

Lesson
Practice
Lesson

Concept summary

An angle is formed wherever two lines, segments or rays intersect. There are two postulates that allow us to measure and solve problems with angles.

Protractor postulate

Consider a ray \overrightarrow{OB} and a point A on one side of \overrightarrow{OB}. Every ray of the form \overrightarrow{OA} can be paired one to one with a real number from 0 to 180. The measure of \angle AOB, written as m \angle AOB, is equal to the difference between the real numbers matched with \overrightarrow{OA} and \overrightarrow{OB} on a protractor.

Angle A O B drawn on a diagram of a protractor. Point O at the center point of the protractor. Ray O A is aligned with the 0 mark, and ray O B is aligned with the 120 mark.
Angle addition postulate

If P is in the interior of angle RST, then m\angle{RSP}+m\angle{PST}=m\angle{RST}.

Adjacent angles R S P and P S T.

The angle addition postulate only works for adjacent angles, defined as angles that share a common leg and vertex, but do not overlap.

The measure of an angle is defined using the protractor postulate.

Angle measure

The amount of rotation needed to map one leg of the angle to the other. Angles can be measured in degrees from 0\degree to 360\degree.

An angle with an angle decoration and label showing that it measures 145 degrees.
Congruent angles

Angles with the same measure. Congruent angles are labeled with arc marks.

Two angles with the same angle decorations each labeled as 57 degrees.

Angles can be classified based on their measure:

Acute angle

An angle whose measure is between 0 and 90 degrees.

An angle of 55 degrees, with a reference line shown at 90 degrees.
Right angle

An angle whose measure is exactly 90 degrees.

An angle formed by a horizontal and vertical ray, labelled as 90 degrees.
Obtuse angle

An angle whose measure is between 90 and 180 degrees.

An angle of 120 degrees, with a reference line shown at 90 degrees.
Straight angle

An angle whose measure is exactly 180 degrees.

An angle formed by rays pointing in opposite directions, labeled as 180 degrees.
Reflex angle

An angle whose measure is between 180 and 360 degrees.

An angle of 210 degrees, with a reference line shown at 180 degrees.

To construct a copy of an angle, we will:

  1. Identify the angle we want to copy.
  2. Draw a ray that will form one of the legs of the copied angle.
  3. With the compass point on the vertex of the original angle, use the compass to draw an arc that intersects both legs.
  4. Copy the arc in Step 3 by placing the point end of the compass onto the endpoint of the ray.
  5. On the original angle, use the compass to measure the distance between the points where the legs of the angle meets the arc drawn in Step 3.
  6. Without changing the compass width, copy the distance by placing the compass point where the ray meets the copied arc and draw an intersecting arc.
  7. Draw a ray that shares its end point with the ray from Step 2, and goes through the intersection found in Step 6.
A diagram showing the 7 steps of constructing a copy of an angle. Speak to your teacher for more information.

Worked examples

Example 1

Determine the measure of the angle being measured by the protractor.

An angle drawn on a diagram of a protractor. The vertex of the angle is at the center point of the protractor. A ray is aligned with the 0 mark, and the other ray is aligned with the 110 mark.

Approach

Using the protractor postulate we can see that one ray of the angle aligns with 0 \degree and the other ray aligns with 110\degree so the measure of the angle is the difference between 0 and 110.

Solution

110\degree

Example 2

Solve for x.

Two angles with the same angle decorations. One angle has a measure of 152 degrees, and the other angle has a measure of x degrees.

Approach

The angles in the diagram are marked as congruent. That means they have equal measure.

Solution

x=152

Example 3

Consider the diagram, where m \angle PQR = 145 \degree.

Adjacent angles P Q S and S Q R. P Q S has a measure of 3 x plus 7 degrees, and S Q R has a measure of 2 x minus 2 degrees.
a

Write an equation and solve for x.

Approach

Using the angle addition postulate we know that m\angle{PQS}+m\angle{SQR}=m\angle{PQR}. Now we can substitute and solve.

Solution

\displaystyle m\angle{PQS}+m\angle{SQR}\displaystyle =\displaystyle m\angle{PQR}Angle addition postulate
\displaystyle 3x+7+2x-2\displaystyle =\displaystyle 145Substitute
\displaystyle 5x+5\displaystyle =\displaystyle 145Combine like terms
\displaystyle 5x\displaystyle =\displaystyle 140Subtract 5 from both sides
\displaystyle x\displaystyle =\displaystyle 28Divide both sides by 5
b

Find m\angle{SQR}.

Approach

Now that we know the value of x we can substitute it back into the expression for m\angle{SQR}.

Solution

\displaystyle m\angle{SQR}\displaystyle =\displaystyle 2x-2
\displaystyle m\angle{SQR}\displaystyle =\displaystyle 2(28)-2Substitute
\displaystyle 2x-2\displaystyle =\displaystyle 54Simplify

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.5.1

Construct a copy of a segment or an angle.

What is Mathspace

About Mathspace