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Middle Years

9.02 Measuring angles

Lesson

An angle is formed between two lines, rays, or segments whenever they intersect. We can think of an angle as a turn from one object to the other.

The most important angle in geometry is called a right angle, and represents a quarter of a turn around a circle. When two objects form a right angle, we say they are perpendicular. We draw a right angle using a small square rather than a circular arc:

Two perpendicular segments.

We draw all other angles with a circular arc. An angle that is smaller than a right angle is called an acute angle. Here are two:

Two right angles together form a straight angle:

Four right angles is the same as two straight angles, making a full revolution:

An angle that is larger than a right angle but smaller than a straight angle is called an obtuse angle. Here are two:

We met the last kind of angle in the previous lesson - a reflex angle is larger than a straight angle, but smaller than a full revolution. Here are two:

Angle types

Angles are a measure of turning. All angles can be compared to a right angle, representing a quarter turn.

Practice question

Question 1

Select the obtuse angle:

  1. Two rays that share a common endpoint having an angle of 180 degrees.

    A

    Two line segments that share a common endpoint having an angle of 90 degrees.

    B

    Two line segments that share a common endpoint having an angle more than 90 degrees but less than 180 degrees.

    C

    Two line segments that share a common endpoint having an angle less than 90 degrees.

    D

Measuring angles

We divide a full revolution up into $360$360 small turns called degrees, and write the unit using a small circle, like this: $360^\circ$360°.

Since $90$90 is one quarter of $360$360, we know that a right angle is exactly $90^\circ$90°. This circle has markings every $45^\circ$45°:

We can measure angles more precisely using a protractor, or an applet like this one:

This lets us associate numbers with the angle types we learned about above.

Angle size

A full revolution is made up of $360$360 degrees, a single degree is written $1^\circ$1°.

Angle type Angle size
Acute angle Larger than $0^\circ$0°, smaller than $90^\circ$90°.
Right angle $90^\circ$90°
Obtuse angle Larger than $90^\circ$90°, smaller than $180^\circ$180°.
Straight angle $180^\circ$180°
Reflex angle Larger than $180^\circ$180°, smaller than $360^\circ$360°.
Full revolution $360^\circ$360°

 

Practice question

Question 2

Select the angle that is closest to $120^\circ$120°:

  1. A small square indicates that the angle is a right angle.

    A

    The angle is close to 135 degrees.

    B

    The angle is close to 135 degrees.

    C

    The angle is greater than 135 degrees but less than 180 degrees.

    D

 

Angle relationships

Whenever two angles share a defining line, ray, or segment, and do not overlap, we say they are adjacent angles. Here are some examples:

Whenever two segments, lines, or rays intersect at a point, two pairs of equal angles are formed. Each angle in the pair is on the opposite side of the intersection point, and they are called vertically opposite angles.

Four angles are formed by the intersecting lines, and there are two pairs of equal angles. Each pair are vertically opposite angles.

If two angles form a right angle, we say they are complementary. We then know that they add to $90^\circ$90°.

If two angles form a straight angle, we say they are supplementary. We then know that they add to $180^\circ$180°.

Whenever we know that two (or more) angles form a right angle, a straight angle, or a full revolution, we can write an equation that expresses this relationship.

Worked example

Solve for the value of $x$x in the diagram below:

Think: The angle formed is a full revolution, so adding these angles all together will make $360^\circ$360°.
Do: We write the equation:

$x+147+116=360$x+147+116=360

We then use subtraction to make $x$x the subject:

$x=360-147-116$x=360147116

We then do the subtraction to find $x$x:

$x=97$x=97

Reflect: We never use degrees once we are working with an equation. We are solving for the value of $x$x, and we don't want to double up on using the degree symbol!

 

Practice question

Question 3

The angles in the diagram below are complementary. What is the value of $x$x?

A right angle, marked by a green box, is divided by a line segment, creating two angles: a top angle and a bottom angle. The top angle, shaded in red, measures x degrees, while the bottom angle, shaded in blue, measures 39 degrees.

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