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1.05 Angles and constructions

Using properties of angles to solve problems

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, or angles that share a common leg and vertex, but do not overlap.

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

Congruent angles

Angles with the same measure.

Two angles with the same angle decorations each labeled as 57 degrees.
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, which means the sum of their angles is 180\degree.

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.

Exploration

Check the box to 'show reflex angle' and drag point A to change the measure of the angle.

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  1. Create the following types of angles and observe the reflex angle: right, acute, obtuse, straight.
  2. What do you notice about each reflex angle?

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 vertical reference ray shown at 90 degrees.
Right angle

An angle whose measure is exactly 90 degrees.

An angle formed by a horizontal and vertical rays, labeled as 90 degrees.
Obtuse angle

An angle whose measure is between 90 and 180 degrees.

An angle of 120 degrees, with a vertical reference ray 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 ray shown at 180 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.

Examples

Example 1

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.
Worked Solution
Create a strategy

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

Apply the idea

x=152

Example 2

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.

Worked Solution
Create a strategy

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

Apply the idea
\displaystyle m\angle{PQS}+m\angle{SQR}\displaystyle =\displaystyle m\angle{PQR}Angle addition postulate
\displaystyle 3x+7+2x-2\displaystyle =\displaystyle 145Substitute m\angle{PQS}=3x+7, m\angle{SQR}=2x-2 and m\angle{PQR}=145
\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}.

Worked Solution
Create a strategy

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

Apply the idea
\displaystyle m\angle{SQR}\displaystyle =\displaystyle 2x-2Expression for m\angle{SQR}
\displaystyle m\angle{SQR}\displaystyle =\displaystyle 2(28)-2Substitute x=28
\displaystyle m\angle{SQR}\displaystyle =\displaystyle 54Evaluate the multiplication and subtraction

Example 3

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.
Worked Solution
Create a strategy

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.

Apply the idea
\displaystyle x+39\displaystyle =\displaystyle 90Definition of complementary angles
\displaystyle x\displaystyle =\displaystyle 51Subtract 39 from both sides
Reflect and check

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

Example 4

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, angle C F D has a measure of 62 degrees, and angle E F D has a measure of 118 degrees.
a

Vertical angles

Worked Solution
Create a strategy

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.

Apply the idea

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

b

Linear pair

Worked Solution
Create a strategy

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.

Apply the idea

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

Reflect and check

There are multiple linear pairs in the diagram, including \angle{EFD} and \angle{CFD}, \angle{AFC} and \angle{CFD}, and \angle{AFC} and \angle{AFE}.

Idea summary

We can use protractors and algebra to measure angles and solve problems involving angles. We can use definitions and postulates for supplementary, complementary, vertical, and adjacent angles to solve problems.

Angle constructions

Exploration

Use the 'Next' arrows to view the construction.

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  1. Describe what is happening at each step of the construction.

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.

Examples

Example 5

Construct a copy of the angle shown.

An angle measuring 135 degrees.
Worked Solution
Create a strategy

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.

Apply the idea
A screenshot of the GeoGebra geometry tool showing an angle measuring 135 degrees, and a separate segment D E. Speak to your teacher for more details.

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 points. 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.

Reflect and check

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 6

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

Circle O with radii O A and O B shown.

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

Worked Solution
Create a strategy

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.

Apply the idea
A screenshot of the GeoGebra geometry tool showing the circle centered at O, the radii O A and O B, and a smaller circle also centered at O along with its points of intersection with the radii. Speak to your teacher for more details.

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}.

Reflect and check

We can also find the midpoint of minor arc \overset{\large\frown}{AB} using string.

First, take a piece of string and tie it to the end of a pencil. Use a push pin to secure the other end of the string on point A and draw an arc.

Circle O with radii O A and O B. A push pin is at A. A string connects the push pin and the pencil. An arc is drawn. Speak to your teacher for more details.

Then, using the same string, place the push pin to secure the string on point B and draw another arc, intersecting the first.

Circle O with radii O A and O B. A push pin is at B. A string connects the push pin and the pencil. Arcs are drawn. Speak to your teacher for more details.

Finally, draw a line from point O through the intersection of the two drawn arcs to the circle's edge to find the midpoint of minor arc \overset{\large\frown}{AB}.

Circle O with radii O A and O B. A point C is on the circle and between A and B. Segment O C is drawn. Speak to your teacher for more details.
Idea summary

A compass and straightedge can be used to construct angles as well as the use of technology or string.

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