12.02 Measuring angles

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.

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.

We met this kind of angle in the previous lesson - a reflex angle which is larger than a straight angle, but smaller than a full revolution.

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

Examples

Example 1

Select the obtuse angle:

A

B

C

D

Worked Solution

Create a strategy

Use this image to help you:

Rotate each angle so that one arm lies over the start.

Apply the idea

The answer is Option C.

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

Since 90 is one quarter of 360, we know that a right angle is exactly 90\degree.

This circle has markings every 45\degree.

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

A full revolution is made up of 360 degrees, a single degree is written as 1\degree.

Examples

Example 2

Select the angle that is closest to 120\degree:

A

B

C

D

Worked Solution

Create a strategy

Use this image to help you:

Rotate each angle so that one arm lies over the start.

Apply the idea

A 120\degree angle would be an obtuse angle between 90\degree and 135\degree.

The answer is Option C.

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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 vertical angles.

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

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

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.

Examples

Example 3

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

Worked Solution

Create a strategy

Complementary angles are two angles forming a right angle equivalent to 90\degree.

Apply the idea

\displaystyle x + 39	\displaystyle =	\displaystyle 90	Equate the sum of the angles to 90
\displaystyle x	\displaystyle =	\displaystyle 51	Subtract 39 from both sides

Reflect and check

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

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