Measuring Angles

An angle is formed when two rays or line segments meet at a common point called the vertex.

Angles describe the amount of turn between two directions and are measured in degrees, written using the symbol °.

For example:

45°

means an angle measuring 45 degrees.

Understanding angles allows us to solve problems involving shapes, intersecting lines, construction, navigation, engineering, and design.

Parts of an Angle

An angle has three important parts:

• vertex — the point where the two rays meet
• arms — the rays or line segments forming the angle
• angle measure — the amount of rotation between the arms

For an angle named ∠ABC:

B is the vertex.

The middle letter always identifies the vertex.

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Measuring Angles

A protractor is commonly used to measure angles.

Most classroom protractors measure from:

0° to 180°

To measure an angle:

1. Place the centre mark of the protractor on the vertex.
2. Align the 0° line with one arm of the angle.
3. Find where the other arm crosses the scale.
4. Read the correct scale.
5. Record the measurement using the degree symbol.

For example:

angle = 63°

Choosing the Correct Protractor Scale

Many protractors have two sets of numbers.

One scale begins:

0°, 10°, 20°, 30°...

from the left.

The other begins:

0°, 10°, 20°, 30°...

from the right.

Always start reading from the 0° aligned with the first arm of the angle.

A useful check is to estimate the angle first.

If the angle clearly looks acute but your protractor gives:

130°

you have probably read the wrong scale.

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Classifying Angles

Angles can be classified according to their size.

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Acute Angles

An acute angle measures:

0° < angle < 90°

Examples:

25°

48°

89°

All are smaller than a right angle.

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Right Angles

A right angle measures exactly:

90°

A small square is often drawn at the vertex to indicate a right angle.

Right angles appear frequently in:

• rectangles
• squares
• buildings
• doors
• floor tiles
• coordinate grids

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Obtuse Angles

An obtuse angle measures:

90° < angle < 180°

Examples:

105°

135°

172°

An obtuse angle is larger than a right angle but smaller than a straight angle.

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Straight Angles

A straight angle measures:

180°

It forms a straight line.

This fact is extremely useful:

angles on a straight line add to 180°

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Reflex Angles

A reflex angle is greater than 180° but less than 360°.

For example:

230°

If the smaller angle between two rays is 130°, the corresponding reflex angle is:

360° − 130° = 230°

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A Full Turn

One complete rotation is:

360°

Therefore:

• quarter turn = 90°
• half turn = 180°
• three-quarter turn = 270°
• full turn = 360°

Complementary Angles

Two angles are complementary when their measures add to:

90°

Therefore:

a + b = 90°

Example:

30° + 60° = 90°

So 30° and 60° are complementary.

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Finding a Missing Complementary Angle

Suppose one angle is:

37°

and the two angles are complementary.

Let the missing angle be x.

x + 37° = 90°

Therefore:

x = 90° − 37°

x = 53°

Check:

37° + 53° = 90°

Correct.

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Supplementary Angles

Two angles are supplementary when their measures add to:

180°

Therefore:

a + b = 180°

Example:

120° + 60° = 180°

So 120° and 60° are supplementary.

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Finding a Missing Supplementary Angle

One angle measures:

128°

Find its supplementary angle.

Let the unknown angle be x.

x + 128° = 180°

Therefore:

x = 180° − 128°

x = 52°

Check:

128° + 52° = 180°

Correct.

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Complementary Versus Supplementary

These terms are easy to confuse.

Remember:

Complementary → 90°

Supplementary → 180°

A useful memory trick:

C comes before S

and:

90 comes before 180

So:

Complementary → 90°

Supplementary → 180°

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Angles on a Straight Line

Angles forming a straight line add to:

180°

Suppose two adjacent angles are:

x° and 72°

Then:

x + 72 = 180

Therefore:

x = 108°

This is the same mathematical relationship as supplementary angles.

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Multiple Angles on a Straight Line

Sometimes more than two angles appear along a straight line.

Suppose three angles are:

35°, x°, and 80°

Because they form a straight line:

35 + x + 80 = 180

Combine known angles:

115 + x = 180

Therefore:

x = 65°

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Angles Around a Point

Angles around a single point add to:

360°

This is because a complete turn is 360°.

Suppose three angles around a point are:

110°, 95°, and x°

Then:

110 + 95 + x = 360

205 + x = 360

Therefore:

x = 155°

Intersecting Lines

When two straight lines cross, they form four angles.

Several important relationships appear automatically.

The angles directly opposite each other are called vertical angles.

Adjacent angles form straight lines and are supplementary.

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Vertical Angles

Vertical angles are opposite angles formed when two lines intersect.

Vertical angles are always equal.

Therefore:

opposite angle = opposite angle

If one angle is:

65°

the angle directly opposite it is also:

65°

This does not require measurement.

It follows from the geometry of intersecting lines.

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Example: Vertical Angles

Two lines intersect.

One angle is 118°.

The angle directly opposite it is:

118°

The two remaining angles are supplementary to 118°.

Therefore:

180° − 118° = 62°

So the four angles are:

118°, 62°, 118°, 62°

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Why Vertical Angles Are Equal

Suppose two lines intersect and create angles A, B, C, and D.

Angles A and B form a straight line:

A + B = 180°

Angles B and C also form a straight line:

B + C = 180°

Therefore:

A + B = B + C

Subtract B from both sides:

A = C

So the opposite angles are equal.

This proves the vertical-angle rule.

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Adjacent Angles

Adjacent angles share:

• a common vertex
• a common arm

For example, if two adjacent angles form a straight line, they add to:

180°

However, adjacent angles do not always add to 180°.

They do so only when their outer arms form a straight line.

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Linear Pairs

Two adjacent angles whose non-common sides form a straight line are called a linear pair.

A linear pair always adds to:

180°

For example:

x + 47 = 180

Therefore:

x = 133°

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Intersecting Lines: A Powerful Pattern

When two lines intersect, you really only need to know one angle.

Suppose one angle is:

40°

Then:

• opposite angle = 40°
• adjacent angle = 180° − 40° = 140°
• other opposite angle = 140°

So the four angles are:

40°, 140°, 40°, 140°

Using Algebra with Vertical Angles

Angle relationships can also produce algebra equations.

Suppose two vertical angles are:

3x + 10

and:

5x − 30

Vertical angles are equal.

Therefore:

3x + 10 = 5x − 30

Add 30:

3x + 40 = 5x

Subtract 3x:

40 = 2x

Therefore:

x = 20

Now calculate the angle:

3(20) + 10 = 70°

Check:

5(20) − 30 = 70°

Both angles are equal.

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Using Algebra with Supplementary Angles

Suppose two angles form a straight line.

Their measures are:

2x + 15

and:

3x + 20

Since they form a straight line:

(2x + 15) + (3x + 20) = 180

Combine like terms:

5x + 35 = 180

Subtract 35:

5x = 145

Divide by 5:

x = 29

Now calculate the angles.

First angle:

2(29) + 15 = 73°

Second angle:

3(29) + 20 = 107°

Check:

73° + 107° = 180°

Correct.

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Choosing the Correct Angle Rule

When solving an unknown angle, first look at the diagram.

Ask:

Do the angles form a right angle?

Use:

total = 90°

Do the angles form a straight line?

Use:

total = 180°

Are the angles opposite each other at intersecting lines?

Use:

vertical angles are equal

Do the angles go completely around a point?

Use:

total = 360°

Choosing the correct relationship is often the most important part of the problem.

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Worked Example: Complementary Angles

Two angles form a right angle.

One measures 26°.

Find x.

x + 26 = 90

x = 90 − 26

x = 64°

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Worked Example: Supplementary Angles

Two angles form a straight line.

One measures 143°.

Find x.

x + 143 = 180

x = 37°

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Worked Example: Vertical Angles

Two lines intersect.

One angle is 76°.

Find the opposite angle.

Vertical angles are equal.

Therefore:

x = 76°

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Worked Example: Intersecting Lines

Two lines intersect.

One angle is 76°.

Find the adjacent angle.

Adjacent angles along a straight line are supplementary.

x + 76 = 180

x = 104°

So the four angles are:

76°, 104°, 76°, 104°

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Worked Example: Angles Around a Point

Four angles around a point are:

90°, 110°, 75°, and x°

Angles around a point total 360°.

90 + 110 + 75 + x = 360

275 + x = 360

x = 85°

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Worked Example: Multi-Step Problem

Two lines intersect.

One angle is:

2x + 20

The opposite angle is:

4x − 40

Since vertical angles are equal:

2x + 20 = 4x − 40

Add 40:

2x + 60 = 4x

Therefore:

60 = 2x

x = 30

Calculate the angle:

2(30) + 20 = 80°

The opposite angle is also:

80°

The adjacent angles are:

180° − 80° = 100°

Therefore all four angles are:

80°, 100°, 80°, 100°

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Real-World Applications of Angles

Angles are important in many areas.

Construction

Builders use angles when constructing:

• roofs
• stairs
• walls
• supports

Engineering

Engineers use angles when designing:

• bridges
• machines
• roads
• structures

Navigation

Directions and bearings depend on angles.

Sports

Angles affect:

• shots
• passes
• trajectories
• rebounds

Art and Design

Angles help create:

• perspective
• patterns
• geometric designs

Estimating Angles

Before measuring an angle, estimate its size.

Compare it with familiar reference angles:

45° → half a right angle

90° → right angle

180° → straight line

360° → full turn

For example, if an angle is slightly larger than a right angle, a reasonable estimate might be:

100°–120°

An answer of 35° would clearly be unreasonable.

Estimation is therefore useful for checking measurements.

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Solving Unknown Angles

A reliable strategy is:

STEP 1 — Identify the relationship

Complementary?

Supplementary?

Vertical?

Straight line?

Around a point?

STEP 2 — Write an equation

Example:

x + 68 = 180

STEP 3 — Solve

x = 112°

STEP 4 — Check

112° + 68° = 180°

STEP 5 — Ask whether the answer makes sense

If the diagram shows an obtuse angle, your answer should normally be greater than 90°.

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Common Mistakes

Reading the Wrong Protractor Scale

Always begin with the scale that starts at 0° on the aligned arm.

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Confusing Acute and Obtuse Angles

Acute:

less than 90°

Obtuse:

greater than 90° but less than 180°

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Complementary Angles Add to 180°

Incorrect.

Complementary angles add to:

90°

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Supplementary Angles Add to 90°

Incorrect.

Supplementary angles add to:

180°

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All Opposite Angles Are Equal

Not necessarily.

The rule applies to vertical angles created by two intersecting lines.

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Vertical Means Up and Down

In the term vertical angles, "vertical" does not mean that the angles must point vertically.

Vertical angles are simply opposite angles formed by intersecting lines.

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Adjacent Angles Are Always Supplementary

Incorrect.

They are supplementary only when their outer arms form a straight line.

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Angles Around a Point Add to 180°

Incorrect.

Angles around a point add to:

360°

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Trusting the Diagram Instead of the Mathematics

Geometry diagrams are not always drawn perfectly to scale.

Use the given information and angle relationships rather than simply judging the picture.

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Check Your Understanding

1. What is an angle?
2. What is the vertex of an angle?
3. What unit is used to measure angles?
4. What instrument is commonly used to measure angles?
5. Classify an angle measuring 42°.
6. Classify an angle measuring 90°.
7. Classify an angle measuring 125°.
8. Classify an angle measuring 180°.
9. Classify an angle measuring 240°.
10. What is the total angle in one complete turn?
11. What do complementary angles add to?
12. Find the complement of 34°.
13. Find the complement of 67°.
14. What do supplementary angles add to?
15. Find the supplement of 115°.
16. Find the supplement of 47°.
17. Two angles form a straight line. One is 68°. Find the other.
18. Three angles on a straight line are 35°, 92°, and x°. Find x.
19. What do angles around a point add to?
20. Three angles around a point are 120°, 85°, and x°. Find x.
21. What are vertical angles?
22. Two lines intersect. One angle is 52°. Find the opposite angle.
23. One angle formed by two intersecting lines is 52°. Find each adjacent angle.
24. Two lines intersect and one angle is 135°. Determine all four angles.
25. Explain why vertical angles are equal.
26. Two vertical angles are 3x + 5 and 5x − 35. Find x.
27. Two supplementary angles are 2x + 10 and 3x + 20. Find x.
28. Explain how you decide which scale to read on a protractor.
29. Why is estimating an angle useful before measuring it?
30. Explain how complementary, supplementary, and vertical-angle relationships can be used to find unknown angles.

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Key Terms

Angle — Amount of rotation or turn between two rays or line segments meeting at a vertex.

Degree (°) — Standard unit used to measure angles.

Vertex — Common endpoint where the arms of an angle meet.

Protractor — Instrument used to measure and draw angles.

Acute angle — Angle greater than 0° and less than 90°.

Right angle — Angle measuring exactly 90°.

Obtuse angle — Angle greater than 90° and less than 180°.

Straight angle — Angle measuring exactly 180°.

Reflex angle — Angle greater than 180° and less than 360°.

Complementary angles — Two angles whose measures add to 90°.

Supplementary angles — Two angles whose measures add to 180°.

Vertical angles — Opposite angles formed by two intersecting lines; they have equal measures.

Adjacent angles — Angles sharing a common vertex and a common arm.

Linear pair — Two adjacent angles whose non-common arms form a straight line; their measures add to 180°.

Intersecting lines — Lines that cross at a common point.

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Key Takeaways

• Angles are measured in degrees (°).
• A protractor can be used to measure angles.
• Estimate an angle before measuring to help identify mistakes.
• Angle classifications include:

acute → less than 90°

right → 90°

obtuse → between 90° and 180°

straight → 180°

reflex → between 180° and 360°

• Complementary angles add to:

90°

• Supplementary angles add to:

180°

• Angles on a straight line add to:

180°

• Angles around a point add to:

360°

• Vertical angles formed by intersecting lines are:

equal

• If one angle formed by intersecting lines is known, all four angles can be determined.
• Algebra can be combined with angle relationships to solve more complex problems.

A useful problem-solving guide is:

LOOK AT THE ANGLE RELATIONSHIP

↓

Right angle? → 90°

Straight line? → 180°

Vertical angles? → EQUAL

Around a point? → 360°

↓

WRITE AN EQUATION

↓

SOLVE FOR THE UNKNOWN

↓

CHECK THE ANGLE RELATIONSHIP

↓

ASK: DOES THE ANSWER MAKE SENSE?