Lesson 1 ~25 min Unit 3 · Geometry +85 XP

Angles, Lines, and Geometry Foundations

Classify angles by size, name angles using vertex notation $\angle ABC$, and use complementary, supplementary, vertically opposite and angles at a point relationships to find unknowns.

Today's hook: A skateboarder does a 360 spin. A surfer carves a 90° turn. Why do we measure turns in degrees, and how do architects use angles every day — from roof pitches to staircase rises?

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Think First

warm-up

Two angles meet at a point on a straight line. One is $50^{\circ}$. What must the other be, and why does that have to be true? Now think about a "+" sign — how many angles are there at the centre, and what do they add up to?

Record your answer in your workbook.

The Big Idea

+5 XP

An angle is the amount of turning between two rays that share a common endpoint (the vertex). We measure angles in degrees (°), where a full turn is $360^{\circ}$ — an idea the ancient Babylonians invented over 4000 years ago.

Angles are classified by their size. An acute angle is less than $90^{\circ}$. A right angle is exactly $90^{\circ}$ and is marked with a small square. An obtuse angle is between $90^{\circ}$ and $180^{\circ}$. A straight angle is exactly $180^{\circ}$, and a reflex angle is more than $180^{\circ}$ but less than $360^{\circ}$.

Acute < 90° < Right = 90° < Obtuse < 180° = Straight < Reflex < 360°

Acute = "a cute" little angle

Small and sharp — less than a right angle.

Right = square corner

Marked with a tiny square. Walls and book corners are right angles.

Reflex = goes the long way

More than a straight line but less than a full turn.

What to write in your book

An angle measures the amount of rotation between two rays meeting at a vertex (measured in degrees).
Classify by size: acute $<90^{\circ}$, right $=90^{\circ}$, obtuse $90^{\circ}–180^{\circ}$, straight $=180^{\circ}$, reflex $180^{\circ}–360^{\circ}$.
In angle name $\angle ABC$ the middle letter ($B$) is always the vertex (the corner).

Quick check — what type of angle measures exactly $90^{\circ}$?

What You'll Master

objectives

Know

Acute < 90°, right = 90°, obtuse 90°–180°, straight = 180°, reflex 180°–360°
Complementary angles sum to 90°; supplementary angles sum to 180°
Vertically opposite angles are equal
Angles at a point sum to 360°

Understand

Why an angle measures the amount of rotation, not the length of the rays
The angle naming convention $\angle ABC$ (vertex letter in the middle)
That straight-line angles and right angles give us "rulers" for finding unknowns

Can Do

Classify any angle as acute, right, obtuse, straight or reflex
Find unknown angles using complementary, supplementary, vertical and point relationships
Justify each step with a geometric reason

Words You Need

vocabulary

VertexThe common endpoint where the two rays of an angle meet. Plural: vertices.

RayA half-line: starts at a point and extends infinitely in one direction.

ComplementaryTwo angles whose measures add to $90^{\circ}$.

SupplementaryTwo angles whose measures add to $180^{\circ}$.

Vertically oppositeThe pair of equal angles formed across the vertex when two lines cross.

Angles at a pointAll angles meeting at one vertex add to $360^{\circ}$.

Spot the Trap

heads-up

Wrong: "$\angle ABC$ means the angle starts at $A$." No — the middle letter is the vertex. $\angle ABC$ has its vertex at $B$.

Right: The middle letter is the corner. $\angle PQR$ and $\angle RQP$ are the same angle — both have vertex $Q$.

Wrong: "Longer rays make a bigger angle." Wrong! Angle size depends only on the rotation, not how far the rays extend.

Right: Complementary = 90 (think "Corner = 90"). Supplementary = 180 (think "Straight = 180").

Complementary and Supplementary Angles

+5 XP

Two angles are complementary if they add to $90^{\circ}$ (they fit perfectly into a right angle). They are supplementary if they add to $180^{\circ}$ (they fit perfectly along a straight line).

If two angles sit side-by-side and form a right angle, they must be complementary. If two angles sit side-by-side and form a straight line, they must be supplementary. To find an unknown, just subtract: complement of $\theta$ is $90 - \theta$; supplement of $\theta$ is $180 - \theta$.

Complementary: $\alpha + \beta = 90^{\circ}$ Supplementary: $\alpha + \beta = 180^{\circ}$

C = Corner

Complementary ↔ Corner (right angle, 90°).

S = Straight

Supplementary ↔ Straight (line, 180°).

Just subtract

Missing complement = $90 - $ given. Missing supplement = $180 - $ given.

What to write in your book

Complementary angles sum to $90^{\circ}$ — "C" for Corner.
Supplementary angles sum to $180^{\circ}$ — "S" for Straight line.
To find an unknown: complement of $\theta = 90 - \theta$; supplement of $\theta = 180 - \theta$.

True or false?

If two angles are supplementary and one of them is $112^{\circ}$, the other must be $68^{\circ}$.

Vertically Opposite & Angles at a Point

+5 XP

When two straight lines cross, four angles form. The angles directly opposite each other (across the vertex) are vertically opposite and they are always equal. All four angles together meet at one point, so they add to $360^{\circ}$.

Two crossing lines form two pairs of vertically opposite angles. Each pair sits on opposite sides of the vertex. Adjacent angles along either line form a straight angle (supplementary, 180°). All four angles at the crossing point together total $360^{\circ}$.

Vertically opposite angles are equal | Angles at a point sum to $360^{\circ}$

"X marks the equal spots"

Two crossing lines make an X — opposite arms are equal.

Always give a reason

Write "(vert. opp. ∠s)" or "(∠s on a str. line)" next to your working.

A full turn = 360°

Any number of angles at the same vertex must add to $360^{\circ}$.

What to write in your book

When two straight lines cross, vertically opposite angles (across the X) are equal.
All angles meeting at a single point sum to $360^{\circ}$.
Always write a reason: (vert. opp. ∠s), (∠s on str. line), (∠s at a pt.).

Fill in the blank.

All the angles meeting at a single point must add up to °.

Watch Me Solve It · 3 examples

Watch Me Solve It · Complementary angle

+15 XP per step

PROBLEM

Find the complement of $34^{\circ}$, then classify the original angle.

1
Recall the definition
Complementary angles sum to $90^{\circ}$
"C" for Complementary, "C" for Corner.
2
Subtract from 90
$90 - 34 = 56^{\circ}$
3
Classify $34^{\circ}$
$34^{\circ} < 90^{\circ}$ → acute
The complement $56^{\circ}$ is also acute — both must be acute to sum to 90.

AnswerComplement $= 56^{\circ}$; $34^{\circ}$ is acute.

Watch Me Solve It · Vertically opposite

+15 XP per step

PROBLEM

Two straight lines cross. One angle is $115^{\circ}$. Find the other three angles.

1
The vertically opposite angle
$115^{\circ}$ (vert. opp. ∠s)
Across the X — always equal.
2
An adjacent angle along the line
$180 - 115 = 65^{\circ}$ (∠s on str. line)
3
The fourth angle
$65^{\circ}$ (vert. opp. to the $65^{\circ}$)
Check: $115 + 65 + 115 + 65 = 360^{\circ}$ ✓

AnswerAngles: $115^{\circ}, 65^{\circ}, 115^{\circ}, 65^{\circ}$

Watch Me Solve It · Angles at a point

+15 XP per step

PROBLEM

Three angles meet at a single point. They measure $120^{\circ}, 90^{\circ}$ and $x^{\circ}$. Find $x$.

1
Set up the equation
$120 + 90 + x = 360$ (∠s at a pt.)
All angles at a single vertex sum to $360^{\circ}$.
2
Simplify the known side
$210 + x = 360$
3
Solve for $x$
$x = 360 - 210 = 150^{\circ}$
This is an obtuse angle.

Answer$x = 150^{\circ}$ (obtuse)

Common Pitfalls

heads-up

Confusing complementary and supplementary

Students often swap the two and use 180° when they should use 90° (or vice versa).

Fix: "C" = Corner = 90°; "S" = Straight = 180°.

Naming the wrong vertex

Writing $\angle ABC$ but thinking the vertex is $A$. The vertex is always the middle letter.

Fix: Underline the middle letter when reading an angle name.

Forgetting to state a reason

An answer without justification loses marks. Every step needs a geometric reason.

Fix: Use shorthand — (vert. opp. ∠s), (co-int. ∠s), (∠s on str. line), (∠s at a pt.).

Copy Into Your Books

Classifying Angles

Acute: <90°
Right: =90°
Obtuse: 90–180°
Straight: =180°; Reflex: 180–360°

Angle Naming

$\angle ABC$ → vertex at $B$
Middle letter = corner
$\angle ABC = \angle CBA$

Angle Relationships

Complementary: sum 90°
Supplementary: sum 180°
Vertically opposite: equal
Angles at a point: sum 360°

Reasons (shorthand)

(vert. opp. ∠s)
(∠s on str. line)
(∠s at a pt.)
(comp. ∠s) / (supp. ∠s)

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Angle Foundations

4 problems

Four drill problems to sharpen your angle skills. Work each, then reveal the answer.

1 Classify the angle $137^{\circ}$.

It is between 90° and 180°.Obtuse
2 Find the supplement of $73^{\circ}$.

Supplement means sum to 180°.$180 - 73 = 107^{\circ}$
3 Two lines cross. One angle is $42^{\circ}$. What is the angle vertically opposite it?

Vertically opposite angles are equal.$42^{\circ}$
4 Four angles at a point are $90^{\circ}, 80^{\circ}, 100^{\circ}, x^{\circ}$. Find $x$.

Sum = 360°.$360 - 270 = 90^{\circ}$

Complete in your workbook.

Quick Check · 5 questions

An angle measures $42^{\circ}$. Classify it.

+10 XP

Find the complement of $28^{\circ}$.

+10 XP

Two lines cross. One angle is $113^{\circ}$. What is the vertically opposite angle?

+10 XP

Three angles meet at a point: $145^{\circ}, 95^{\circ}, x^{\circ}$. Find $x$.

+10 XP

In the notation $\angle PQR$, which point is the vertex?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Classify each angle as acute, right, obtuse, straight or reflex: (a) $89^{\circ}$, (b) $180^{\circ}$, (c) $217^{\circ}$, (d) $90^{\circ}$, (e) $120^{\circ}$, (f) $1^{\circ}$.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. Find the value of $x$, giving a reason for each step:
(a) $x$ and $35^{\circ}$ are complementary.
(b) $x$ and $112^{\circ}$ are supplementary.
(c) Three angles at a point are $90^{\circ}, 130^{\circ}$ and $x^{\circ}$.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. Two straight lines cross at a point $O$. One of the angles is $4y^{\circ}$ and its supplementary neighbour along one line is $(2y + 30)^{\circ}$. Find $y$ and state the size of every angle at $O$.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — Acute. $42^{\circ} < 90^{\circ}$.

2. A — $62^{\circ}$. Complement = $90 - 28$.

3. D — $113^{\circ}$. Vertically opposite angles are equal.

4. B — $120^{\circ}$. $360 - 145 - 95 = 120$.

5. C — $Q$. Middle letter is the vertex.

Show Your Working Model Answers

Q6 (3 marks): (a) acute, (b) straight, (c) reflex, (d) right, (e) obtuse, (f) acute. [1 mark per 2 correct]

Q7 (3 marks): (a) $x = 90 - 35 = 55^{\circ}$ (comp. ∠s) [1]. (b) $x = 180 - 112 = 68^{\circ}$ (supp. ∠s) [1]. (c) $x = 360 - 90 - 130 = 140^{\circ}$ (∠s at a pt.) [1].

Q8 (3 marks): Angles on a line: $4y + (2y + 30) = 180$ [1]. $6y = 150$, so $y = 25$ [1]. Angles: $4y = 100^{\circ}$ and $2y + 30 = 80^{\circ}$, with their vertically opposite partners $100^{\circ}$ and $80^{\circ}$ [1].

Stretch Challenge · +25 XP, +10 coins

The Clockface Puzzle

At exactly 3:00, the hour and minute hands of a clock form a right angle. What angle do they form at 4:00? At 5:30? Justify each answer by considering how many "12ths of $360^{\circ}$" each hand has turned.

Reveal solution

Each hour gap = $360 \div 12 = 30^{\circ}$. At 4:00 the hands are 4 gaps apart = $120^{\circ}$ (obtuse). At 5:30 the minute hand is on 6 and the hour hand is halfway between 5 and 6, so they are $0.5 \times 30 = 15^{\circ}$ apart (acute).

Quick Review

Acute

Less than $90^{\circ}$ — sharp.

Right / Obtuse

Right = $90^{\circ}$; Obtuse = 90°–180°.

Straight / Reflex

Straight = $180^{\circ}$; Reflex = 180°–360°.

Complement

Two angles summing to $90^{\circ}$.

Supplement

Two angles summing to $180^{\circ}$.

At a point

All angles at one vertex sum to $360^{\circ}$.

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