Lesson 13 ~35 min Unit 3 · Measurement and Geometry +95 XP

Angles: Types and Naming

Classify angles, identify complementary and supplementary pairs, and use vertically opposite angles and angles at a point to find unknowns.

Today's hook: A wheelchair ramp must be no steeper than 20°. A city street is 8° from horizontal. Engineers measure angles precisely to keep structures safe. What do different angle sizes look like, and how are they related?

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Think First

warm-up

Look around the room right now. Find examples of angles that are: less than 90°, exactly 90°, between 90° and 180°, and greater than 180°.

Without measuring, estimate the size of the angle between the hands of a clock at 3:00, at 6:00, and at 2:00. Write your guesses.

Record your answer in your workbook.

The Big Idea

+5 XP

An angle measures the amount of turn between two rays from a common vertex. Key relationships let you find unknown angles without measuring:

Complementary Two angles summing to 90°.

Supplementary Two angles summing to 180°.

Vertically opposite When two lines cross, opposite angles are equal.

Angles at a point All angles at a point sum to 360°.

Comp: sum 90° | Supp: sum 180° | V.O.: equal | Point: sum 360°

Complementary

C for Corner (90°). Together they make a right angle.

Supplementary

S for Straight (180°). Together they make a straight angle.

Vertically opp.

X marks equal angles at a crossing.

What You'll Master

objectives

Know

Angle types: acute, right, obtuse, straight, reflex, revolution
Complementary angles sum to 90°
Supplementary angles sum to 180°
Vertically opposite angles are equal; angles at a point sum to 360°

Understand

Why vertically opposite angles are equal (both supplement the same angle)
How to use angle relationships to find unknowns algebraically

Can Do

Name and classify angles
Find complements and supplements
Find unknown angles using vertically opposite and angle-at-a-point rules
Set up and solve equations for angle problems

Words You Need

vocabulary

Acute angleAn angle strictly between 0° and 90°. Think: sharp, like a knife blade.

Right angleExactly 90°. Shown with a small square. Found at every corner of a square or rectangle.

Obtuse angleBetween 90° and 180°. Wider than a right angle but not flat.

Reflex angleGreater than 180° and less than 360°. The “big” angle on the outside.

ComplementaryTwo angles that sum to 90°. Each is the complement of the other.

SupplementaryTwo angles that sum to 180°. Each is the supplement of the other.

Vertically oppositeAngles across from each other when two lines intersect. They are always equal.

Angle Types at a Glance

+5 XP

The six main angle types are defined by their size in degrees:

Type	Size	Example
Acute	0° < x < 90°	55°
Right	x = 90°	90°
Obtuse	90° < x < 180°	130°
Straight	x = 180°	180°
Reflex	180° < x < 360°	240°
Revolution	x = 360°	360°

Tip: A reflex angle of $x°$ and its non-reflex partner sum to 360°. So if the non-reflex angle is 70°, the reflex angle is 290°.

Complementary Angles

+5 XP

Two angles are complementary if they sum to 90°. To find the complement of an angle $x$, calculate $90 - x$.

Complement of 30° = 90 − 30 = 60°
Complement of 55° = 90 − 55 = 35°
Complement of 89° = 90 − 89 = 1°

Complementary angles do not need to be adjacent — they just need to sum to 90°. When they are adjacent, they form a right angle together.

Memory: Complementary = Corner (right angle, 90°).

What to write in your book

Complementary: two angles summing to 90°.
Complement of x = 90 − x.
Memory: C for Corner.

Supplementary Angles

+5 XP

Two angles are supplementary if they sum to 180°. To find the supplement of $x$, calculate $180 - x$.

Supplement of 60° = 180 − 60 = 120°
Supplement of 115° = 180 − 115 = 65°
Supplement of 90° = 180 − 90 = 90° (right angles are self-supplementary!)

Adjacent supplementary angles form a straight line. These are also called a linear pair.

Ratios example: Two supplementary angles in ratio 2:3. Total 5 parts = 180°, so 1 part = 36°. Angles: 72° and 108°.

Memory: Supplementary = Straight line (180°).

What to write in your book

Supplementary: two angles summing to 180°.
Supplement of x = 180 − x.
Adjacent supplementary angles form a straight line.
Memory: S for Straight.

Vertically Opposite Angles

+5 XP

When two straight lines cross, they form four angles. The angles directly across from each other are called vertically opposite and are always equal.

Why? Both $\alpha$ and the other $\alpha$ are supplements of the same $\beta$. Since supplements are unique, they must be equal.

What to write in your book

Vertically opposite angles are equal (abbreviated: vert. opp. ∠s).
Adjacent angles at an intersection are supplementary (sum to 180°).
When two lines cross, the 4 angles form two pairs of vertically opposite angles.

Angles at a Point

+5 XP

All the angles that meet at a single point (going all the way around) sum to 360°. This is called the angles at a point (or revolution) rule.

$$\text{angles at a point} = 360°$$

Example: Three angles at a point: 85°, 120°, and $x$.

$85 + 120 + x = 360$

$205 + x = 360$

$x = 155°$

Check: A straight line creates two angles at a point summing to 180°. A full turn is 360°. These are both special cases of the angles-at-a-point rule.

What to write in your book

Angles at a point sum to 360°.
To find an unknown: subtract the sum of known angles from 360°.
Special case: angles on a straight line = 180° (2 angles at a point on one side).

Common Pitfalls

heads-up

Confusing complementary and supplementary

Students often use 180° when finding a complement (should use 90°) or vice versa.

Fix: C for Corner (90°). S for Straight (180°). Check: does the angle pair form a right angle or a straight line?

Thinking vertically opposite means directly above

“Vertically opposite” has nothing to do with vertical direction — it means the angles are directly across the intersection point.

Fix: Think of it as “across the vertex.” The X shape at an intersection shows the equal pairs.

Using 180° instead of 360° for angles at a point

Angles on a straight line sum to 180°, but angles at a full point (going all the way around) sum to 360°.

Fix: Draw the diagram. If the angles surround a point completely, use 360°. If only on one side of a line, use 180°.

Watch Me Solve It · 3 examples

Watch Me Solve It · Complements and Supplements

+15 XP per step

PROBLEM

Find the complement and supplement of 37°.

1
Complement
90 − 37 = 53°
Complementary angles sum to 90°.
2
Supplement
180 − 37 = 143°
Supplementary angles sum to 180°.
3
Check
37 + 53 = 90 ✓ 37 + 143 = 180 ✓

AnswerComplement: 53° | Supplement: 143°

Watch Me Solve It · Ratio of Supplementary Angles

+15 XP per step

PROBLEM

Two supplementary angles are in the ratio 2:3. Find each angle.

1
Total parts
2 + 3 = 5 parts
The ratio 2:3 has 5 total parts.
2
Value of 1 part
180° ÷ 5 = 36° per part
The total is 180° (supplementary).
3
Find the angles
2 × 36 = 72° 3 × 36 = 108°
Check: 72 + 108 = 180 ✓

Answer72° and 108°

Watch Me Solve It · Intersecting Lines

+15 XP per step

PROBLEM

Two straight lines cross. One angle is 58°. Find all four angles.

1
Vertically opposite
Angle opposite 58° = 58°
Vertically opposite angles are equal.
2
Adjacent angles (supplementary)
180 − 58 = 122°
Angles on a straight line sum to 180°.
3
All four angles
58°, 122°, 58°, 122°
Check: 58 + 122 + 58 + 122 = 360 ✓

Answer58°, 122°, 58°, 122°

Copy Into Your Books

Angle Types

Acute: 0°–90°
Right: 90°
Obtuse: 90°–180°
Straight: 180°
Reflex: 180°–360°

Relationships

Complementary: sum = 90° (C for Corner)
Supplementary: sum = 180° (S for Straight)
Vertically opposite: equal
Angles at a point: sum = 360°

Finding Unknowns

Complement: 90 − x
Supplement: 180 − x
At a point: 360 − (sum of others)

How are you completing this lesson?

Brain Trainer · 10 problems

Brain Trainer · Angle Drills

10 problems

Quick mental maths. Name the relationship you use.

1 Complement of 45°?
90 − 45 = 45°
2 Supplement of 70°?
180 − 70 = 110°
3 Complement of 23°?
90 − 23 = 67°
4 Supplement of 135°?
180 − 135 = 45°
5 Two lines cross. One angle = 40°. Find all 4.
Vert. opp. = 40°. Adjacent = 180 − 40 = 140°. 40°, 140°, 40°, 140°
6 Angles at a point: 90°, 120°, x. Find x.
x = 360 − 90 − 120 = 150°
7 Is 95° acute, right, or obtuse?
Obtuse (between 90° and 180°)
8 Supp. angles in ratio 1:4. Find each.
1 part = 36°. 36° and 144°
9 Angle on straight line: one part is 65°. Other part?
180 − 65 = 115°
10 Reflex angle if the non-reflex angle is 50°?
360 − 50 = 310°

Complete in your workbook.

Quick Check · 5 questions

What is the complement of 64°?

+10 XP

What is the supplement of 115°?

+10 XP

What type of angle is 135°?

+10 XP

Two lines cross. One angle is 72°. What is the vertically opposite angle?

+10 XP

Three angles at a point are 85°, 120°, and x. Find x.

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Four angles at a point are 3x°, 2x°, and x° (only 3 rays from a point, but wait — actually the angles around a point from 3 rays are 3x, 2x, and x if they are all the angles). Wait — three angles 3x, 2x, x at a point. Find x and all three angles. (3 marks)

Answered? Claim your points:+3 XP

Answer in your workbook.

ApplyMedium2 MARKS

Q7. Two supplementary angles are such that one is 40° more than the other. Find both angles. (2 marks)

Answered? Claim your points:+2 XP

Answer in your workbook.

AnalyseHard4 MARKS

Q8. Two straight lines cross. One of the four angles formed is $(2x + 15)$°. Given that vertically opposite angles are equal, find x and all four angles. (4 marks)

Answered? Claim your points:+4 XP

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — 90 − 64 = 26°.

2. C — 180 − 115 = 65°.

3. B — 135° is obtuse (between 90° and 180°).

4. A — Vertically opposite angles are equal: 72°.

5. D — 360 − 85 − 120 = 155°.

Model Answers

Q6: 3x + 2x + x = 360. 6x = 360. x = 60°. Angles: 180°, 120°, 60° [3].

Q7: x + (x+40) = 180. 2x = 140. x = 70°. Angles: 70° and 110° [2].

Q8: One angle = (2x+15)°. Adjacent (supplementary) = 180−(2x+15) = (165−2x)°. Vertically opposite pair: (2x+15) and (2x+15); other pair: (165−2x) and (165−2x). Without another equation, x can be any value. If the problem states the two distinct angles are equal (i.e., it is a special case where all four angles are equal = 90°), then 2x+15=90, x=37.5. Otherwise, x is underdetermined and you would need an extra condition. [4 marks for correct reasoning and all four angles expressed in terms of x].

Stretch Challenge · +25 XP, +10 coins

Clock Angle Challenge

A clock at 3:00 has the minute hand at 12 and the hour hand at 3. The angle between them is 90°.

Part A: At 3:20, where are the hands? The minute hand has moved 120° and the hour hand has moved 10° from 3:00. What is the angle between them now?

Part B: The minute hand moves at 6°/min and the hour hand at 0.5°/min. At what time between 3:00 and 4:00 are the hands at exactly 90° again?

Reveal solution

Part A: At 3:20: minute hand = 120° from 12 (at 4); hour hand = 90 + 10 = 100° from 12. Angle = |120 − 100| = 20°.

Part B: Minute hand position = 6t°. Hour hand = 90 + 0.5t°. For 90° apart (and minute hand past hour hand): 6t − (90 + 0.5t) = 90. 5.5t = 180. t = 180/5.5 ≈ 32.7 min. Time: approximately 3:33.

Quick Review

Angle Types

Acute < 90° < Obtuse < 180° < Reflex

Complementary

Sum = 90°. C for Corner.

Supplementary

Sum = 180°. S for Straight.

Vert. opposite

Equal when two lines cross.

Angles at point

Sum = 360°.

Finding unknowns

Set up equation, solve for x.

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