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8.03 Angles

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.
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.

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.

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

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

Example 4

Construct a copy of the angle shown.

An angle measuring 135 degrees

Approach

To construct a copy of an angle, we will follow the series of steps detailed in the concept summary above, using technology in the form of Geogebra.

Solution

A screenshot of the GeoGebra geometry tool showing an angle measuring 135 degrees, and a separate segment DE.

We start by drawing a ray nearby that will form one leg of the angle. We can do this by using the Vector tool as shown.

A screenshot of the GeoGebra geometry tool showing the previous image with circles drawn around two of the poitns. Speak to your teacher for more details.

Next we want to create an arc on the original angle that intersects both legs, and then duplicate an arc with the same radius centered on the new ray.

To do so, we can make use of the Circle: Center & Radius tool, to ensure that both arcs have the same radius.

A screenshot of the GeoGebra geometry tool showing the previous image with three points of intersection labeled. Speak to your teacher for more details.

We can then use the Point tool to create points at all of the intersections between an arc/circle and a ray.

A screenshot of the GeoGebra geometry tool showing the previous image with a circle drawn centered at one of the points of intersection. Speak to your teacher for more details.

We now want to get the radius of the circle centered at a point of intersection on the original angle (point F in this case) that passes through the other point of intersection (point G).

Once again, we can make use of the Circle: Center & Radius tool, by clicking on F (to be the center) and then dragging to G before releasing. Once we have created the circle, we can check its radius in the algebra tab.

A screenshot of the GeoGebra geometry tool showing the previous image with an additional circle drawn centered at one of the other points of intersection. Speak to your teacher for more details.

We can then duplicate this circle across to the new ray, centering it on the point of intersection (point H) and giving it the same radius using the Circle: Center & Radius tool.

A screenshot of the GeoGebra geometry tool showing the previous image with a new segment joining point D and a new point of intersection. Speak to your teacher for more details.

Finally, we can mark the point of intersection of the two circles we have created, and then use the Vector tool to create the other ray of our copied angle.

Reflection

Note that the new copy of the angle doesn't have to be drawn in the same orientation as the original angle - it can be rotated, as shown here.

Also note that because we have used a circle tool to draw the arcs, there are two possible points of intersection to use at the last step. Using either of these will create an angle of the correct measure.

Example 5

A circle centered at O has radii \overline{AO} and \overline{BO} as shown.

A circle centered at point O with radii OA and OB shown.

Find and label point C, which lies at the midpoint of minor arc \overset{\large\frown}{AB}.

Approach

The ray which bisects \angle{AOB} will intersect the circle at the midpoint of \overset{\large\frown}{AB}. So we can construct an angle bisector and then mark the point of intersection with the circle.

We will do so by making use of technology in the form of Geogebra.

Solution

A screenshot of the GeoGebra geometry tool showing the circle centered at O, the radii OA and OB, and a smaller circle also centered at O along with its points of intersection with the radii.

We start by drawing an arc centered at O that intersects both radii. The easiest way to do this is to use one of the Circle tools.

We then mark both of the points of intersection, using the point tool.

A screenshot of the GeoGebra geometry tool showing the previous image with a small circle centered at one of the points of intersection. Speak to your teacher for more details.

Next we draw an arc centered at one of these points of intersection that passes halfway between the interior of \angle{AOB}.

It is easiest to do this using the Circle: Center & Radius tool, since the next step will be to create another arc (i.e. circle) of the same radius.

A screenshot of the GeoGebra geometry tool showing the previous image with another small circle centered at the other of the point of intersection. The point of intersection of the two new circles is shown. Speak to your teacher for more details.

We now repeat this, creating another circle of the same radius centered at the other point of intersection, so that it crosses the circle we just drew.

We then mark the newly formed point of intersection.

A screenshot of the GeoGebra geometry tool showing the previous image with a line drawn through the most recent point of intersection and point O. Speak to your teacher for more details.

We now construct the bisector of \angle{AOB} by drawing a line through this latest point of intersection and O.

A screenshot of the GeoGebra geometry tool showing the previous image with point C shown as the point of intersection of the new line and the original circle. Speak to your teacher for more details.

Finally, we can label point C as the intersection of the bisector and \overset{\large\frown}{AB}.

Reflection

Notice that the line which bisects \angle{AOB} bisects both the minor arc \overset{\large\frown}{AB} and the major arc on the other side.

So we could use exactly the same process to find the midpoint of a major arc as well, shown as point D in the following diagram:

A screenshot of the GeoGebra geometry tool showing the previous image with point D shown as the point of intersection of the new line and the original circle on the far side of the circle. Speak to your teacher for more details.

Outcomes

M1.G.CO.B.3

Use definitions and theorems about lines and angles to solve problems and to justify relationships in geometric figures.

M1.G.CO.C.5

Perform formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

M1.G.CO.C.6

Use geometric constructions to solve geometric problems in context, by hand and using technology.*

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

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