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

Types of Triangles by Their Angles

Classify triangles as acute-angled, right-angled or obtuse-angled. Identify the hypotenuse in a right triangle, and combine angle-classification with side-classification (e.g. "right isosceles").

Today's hook: Can a triangle have TWO right angles? Think carefully before you answer… If two angles in a triangle were each $90^{\circ}$, what would happen to the third? Try to draw one if you don't believe me!

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Think First

warm-up

The angles in any triangle add to $180^{\circ}$. If one angle is exactly $90^{\circ}$, what must be true about the other two angles? Could you have a triangle with TWO angles each bigger than $90^{\circ}$? Sketch one if you think you can.

Record your answer in your workbook.

The Big Idea

+5 XP

Every triangle can also be classified by its angles. An acute-angled triangle has ALL three angles less than $90^{\circ}$. A right-angled triangle has exactly one angle equal to $90^{\circ}$. An obtuse-angled triangle has exactly one angle greater than $90^{\circ}$. A triangle can have at most one right or obtuse angle — because the three angles must add to $180^{\circ}$.

An acute-angled triangle has every angle below $90^{\circ}$. A right-angled triangle has exactly one $90^{\circ}$ angle, marked with a small square; the side opposite this right angle is called the hypotenuse and is always the longest side. An obtuse-angled triangle has exactly one angle that is bigger than $90^{\circ}$ (but less than $180^{\circ}$).

Acute: all <90° | Right: one =90° | Obtuse: one >90°

Acute = ALL sharp

Every single angle must be less than $90^{\circ}$, not just one or two.

Look for the square

A small square inside an angle is the symbol for $90^{\circ}$. That side opposite it is the hypotenuse.

At most ONE big angle

A triangle can never have two right angles or two obtuse angles — they'd already sum to ≥$180^{\circ}$.

What to write in your book

Acute-angled triangle: ALL three angles are $< 90^{\circ}$.
Right-angled triangle: exactly one $90^{\circ}$ angle (marked with a small square); the side opposite is the hypotenuse and is the longest side.
Obtuse-angled triangle: exactly one angle $> 90^{\circ}$ (and $< 180^{\circ}$).

Quick check — in a right-angled triangle, which side is the hypotenuse?

What You'll Master

objectives

Know

Acute-angled = ALL three angles less than $90^{\circ}$
Right-angled = exactly one angle of $90^{\circ}$; the hypotenuse is the side opposite
Obtuse-angled = exactly one angle greater than $90^{\circ}$
The hypotenuse is always the longest side of a right triangle

Understand

Why a triangle can have at most one right or obtuse angle (angle sum $= 180^{\circ}$)
How angle-classification combines with side-classification (e.g. "right isosceles")
Why the hypotenuse must be the longest side

Can Do

Classify a triangle as acute, right or obtuse from its angles
Identify the hypotenuse in a right-angled triangle
Give a combined classification using both sides and angles

Words You Need

vocabulary

Acute-angledA triangle in which all three angles measure less than $90^{\circ}$.

Right-angledA triangle with exactly one $90^{\circ}$ angle (marked with a small square).

Obtuse-angledA triangle with exactly one angle greater than $90^{\circ}$ but less than $180^{\circ}$.

HypotenuseThe side opposite the right angle in a right triangle — always the longest side.

Right isoscelesA triangle that is both right-angled AND isosceles. Its angles are $45^{\circ}, 45^{\circ}, 90^{\circ}$.

Acute isoscelesA triangle with two equal sides AND every angle below $90^{\circ}$.

Spot the Trap

heads-up

Wrong: "A triangle with one $40^{\circ}$ angle is acute." Not enough info! The OTHER two angles might add up to give an obtuse or right angle.

Right: Acute means ALL THREE angles are less than $90^{\circ}$. Check every angle, not just one.

Wrong: "The hypotenuse is the bottom side of a triangle." No — the hypotenuse is the side opposite the right angle, no matter how the triangle is rotated on the page.

Right: Find the square mark, then look across the triangle — that's the hypotenuse, the longest side.

Why at Most One Right (or Obtuse) Angle

+5 XP

The three angles in a triangle always add to $180^{\circ}$. So if you tried to put two right angles inside, you'd already be at $90 + 90 = 180^{\circ}$ — leaving $0^{\circ}$ for the third corner. Two right angles can't share a triangle. The same is true for two obtuse angles (they'd add to over $180^{\circ}$). So a triangle has, at most, ONE right or obtuse angle.

Suppose two angles were each $90^{\circ}$. Then their sum is already $180^{\circ}$, and there's nothing left for the third angle — the triangle closes up into a single line. The same logic blocks two obtuse angles ($> 90^{\circ}$ each would sum to $> 180^{\circ}$). The third angle, by contrast, can be anything that finishes the sum to $180^{\circ}$.

Angle sum in a triangle $= 180^{\circ}$ ⇒ at most ONE angle $\ge 90^{\circ}$

Two big angles? Impossible

No triangle has two right angles or two obtuse angles. Ever.

Other two are always acute

In a right or obtuse triangle, the OTHER two angles must both be acute (less than $90^{\circ}$).

Sum check

Always check: three angles add to $180^{\circ}$? If not, it's not a valid triangle.

What to write in your book

Angle sum in any triangle is $180^{\circ}$ — this caps the total.
A triangle can have AT MOST ONE right angle or obtuse angle (two would already use $\ge 180^{\circ}$).
In a right or obtuse triangle, the OTHER two angles are always acute.

True or false?

A triangle can have two right angles.

Combining Sides & Angles

+5 XP

You can describe the SAME triangle both ways. A triangle with angles $45^{\circ}, 45^{\circ}, 90^{\circ}$ is right-angled (by its angles) AND isosceles (the two $45^{\circ}$ angles sit opposite two equal sides). So we call it a right isosceles triangle. Common combinations include: right scalene, right isosceles, obtuse scalene, obtuse isosceles, acute scalene, acute isosceles, and acute equilateral (which is the only kind of equilateral).

Every equilateral triangle is acute — its three $60^{\circ}$ angles are all below $90^{\circ}$. An isosceles triangle can be acute, right or obtuse, depending on the size of the apex angle. A scalene triangle can also be acute, right or obtuse. So when classifying, give BOTH descriptors when you can.

Side-name + angle-name → full classification (e.g. "right isosceles")

All equilateral are acute

$60^{\circ} < 90^{\circ}$, so no equilateral can be right or obtuse.

Isosceles ≠ right

An isosceles triangle is right ONLY when the apex angle is $90^{\circ}$. Otherwise it's acute or obtuse.

Two-word names are common

"Right scalene", "obtuse isosceles" etc. are completely normal mathematical descriptions.

What to write in your book

Triangles can carry TWO labels: a side-name (scalene/isosceles/equilateral) AND an angle-name (acute/right/obtuse).
A right isosceles triangle has angles $45^{\circ}, 45^{\circ}, 90^{\circ}$.
Every equilateral triangle is automatically acute (its $60^{\circ}$ angles are all $< 90^{\circ}$).

Fill in the blank.

A right isosceles triangle has two equal angles, each measuring °.

Watch Me Solve It · 3 examples

Watch Me Solve It · Classify by angles

+15 XP per step

PROBLEM

A triangle has angles $40^{\circ}, 60^{\circ}$ and $80^{\circ}$. Classify it by its angles.

1
Check the angle sum
$40 + 60 + 80 = 180^{\circ}$ ✓
Valid triangle.
2
Test each angle against $90^{\circ}$
$40 < 90$, $60 < 90$, $80 < 90$
All three are less than a right angle.
3
Name the type
All angles $< 90^{\circ}$ → acute-angled

AnswerAcute-angled triangle.

Watch Me Solve It · Identify the hypotenuse

+15 XP per step

PROBLEM

A right-angled triangle has sides $6\text{ cm}, 8\text{ cm}$ and $10\text{ cm}$. The right angle is between the $6\text{ cm}$ and $8\text{ cm}$ sides. Which side is the hypotenuse, and why?

1
Locate the right angle
The right angle is between the $6\text{ cm}$ and $8\text{ cm}$ sides.
2
Look across (opposite) the right angle
The side opposite is the $10\text{ cm}$ side
Hypotenuse is always opposite the right angle.
3
Check it's the longest
$10 > 8 > 6$ ✓
The hypotenuse must be the longest side.

AnswerThe $10\text{ cm}$ side is the hypotenuse.

Watch Me Solve It · Combined classification

+15 XP per step

PROBLEM

A triangle has angles $30^{\circ}, 30^{\circ}$ and $120^{\circ}$. Classify it using BOTH its sides and its angles.

1
Angle classification
$120^{\circ} > 90^{\circ}$ → one obtuse angle
Triangle is obtuse-angled.
2
Side classification (from equal angles)
Two equal angles $30^{\circ}, 30^{\circ}$ → the sides opposite them are equal
Equal angles ⇒ equal opposite sides → isosceles.
3
Combine
Both labels: obtuse isosceles

AnswerObtuse isosceles triangle.

Common Pitfalls

heads-up

Calling a triangle "acute" because one angle is acute

In an obtuse triangle, two of the three angles are still acute — that doesn't make the whole triangle acute.

Fix: Acute means ALL three angles are less than $90^{\circ}$. Check every angle.

Putting the hypotenuse next to the right angle

Students sometimes label a side touching the right angle as the hypotenuse.

Fix: Hypotenuse is OPPOSITE the right angle — it never touches the right-angle vertex.

Thinking a triangle can have two right angles

"$90^{\circ} + 90^{\circ} = 180^{\circ}$, that's a triangle's sum, so it works." Wrong — the third angle would have to be $0^{\circ}$.

Fix: A triangle has at most one right or obtuse angle, full stop.

Copy Into Your Books

Acute-angled

ALL angles < $90^{\circ}$
Equilateral $60$° is always acute

Right-angled

Exactly one $90^{\circ}$ angle
Hypotenuse: opposite the right angle
Hypotenuse = longest side

Obtuse-angled

Exactly one angle > $90^{\circ}$
Other two angles are acute

Combined names

Right isosceles ($45,45,90$)
Right scalene
Obtuse isosceles / scalene
Acute equilateral / isosceles

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Angle Classification

4 problems

Four quick drills on the angle types. Solve first, then reveal.

1 Classify the triangle with angles $25^{\circ}, 65^{\circ}, 90^{\circ}$.

One angle is exactly $90^{\circ}$.Right-angled
2 Classify the triangle with angles $20^{\circ}, 30^{\circ}, 130^{\circ}$.

$130^{\circ}$ is greater than $90^{\circ}$.Obtuse-angled
3 In a right triangle, two of the angles are $90^{\circ}$ and $35^{\circ}$. What is the third?

Three angles sum to $180^{\circ}$.$180 - 90 - 35 = 55^{\circ}$
4 A triangle has angles $45^{\circ}, 45^{\circ}, 90^{\circ}$. Give its FULL classification (sides AND angles).

Two equal angles → isosceles; one $90^{\circ}$ → right.Right isosceles

Complete in your workbook.

Quick Check · 5 questions

A triangle has angles $50^{\circ}, 60^{\circ}, 70^{\circ}$. Classify it.

+10 XP

A triangle has angles $40^{\circ}, 25^{\circ}, 115^{\circ}$. Classify it.

+10 XP

In a right-angled triangle, which side is the hypotenuse?

+10 XP

A triangle has angles $45^{\circ}, 45^{\circ}, 90^{\circ}$. Its FULL classification is:

+10 XP

Can a triangle have two right angles?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Easy 3 MARKS

Q6. Classify each triangle by its angles: (a) $30^{\circ}, 60^{\circ}, 90^{\circ}$, (b) $80^{\circ}, 50^{\circ}, 50^{\circ}$, (c) $100^{\circ}, 40^{\circ}, 40^{\circ}$, (d) $60^{\circ}, 60^{\circ}, 60^{\circ}$, (e) $25^{\circ}, 30^{\circ}, 125^{\circ}$, (f) $90^{\circ}, 45^{\circ}, 45^{\circ}$.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. Give the FULL classification (both sides AND angles) for each:
(a) Triangle with angles $60^{\circ}, 60^{\circ}, 60^{\circ}$.
(b) Triangle with angles $90^{\circ}, 45^{\circ}, 45^{\circ}$.
(c) Triangle with angles $110^{\circ}, 40^{\circ}, 30^{\circ}$.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. Explain, using the angle-sum of a triangle, why a triangle cannot have both a right angle AND an obtuse angle. Use a calculation as part of your reasoning.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — Acute-angled. $50, 60, 70$ all less than $90^{\circ}$.

2. C — Obtuse-angled. $115^{\circ} > 90^{\circ}$.

3. D — Side opposite the right angle (the longest side).

4. B — Right isosceles. Two $45^{\circ}$ angles (isosceles) plus one $90^{\circ}$ (right).

5. C — No. $90 + 90 = 180^{\circ}$ leaves $0^{\circ}$ for the third angle.

Show Your Working Model Answers

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

Q7 (3 marks): (a) Equilateral & acute — "acute equilateral" [1]. (b) Two equal angles → isosceles, one $90^{\circ}$ → right — "right isosceles" [1]. (c) All angles different → scalene; $110^{\circ} > 90^{\circ}$ → obtuse — "obtuse scalene" [1].

Q8 (3 marks): Suppose a triangle has a right angle ($90^{\circ}$) AND an obtuse angle ($> 90^{\circ}$) [1]. Their sum is already $> 90 + 90 = 180^{\circ}$ [1]. But the three angles of a triangle must sum to exactly $180^{\circ}$, so there is no room for a third (positive) angle — contradiction [1].

Stretch Challenge · +25 XP, +10 coins

The Six-Category Triangle Map

There are six "combined" categories: acute equilateral, acute isosceles, acute scalene, right isosceles, right scalene, obtuse isosceles, obtuse scalene. Wait — that's seven… or is it? For EACH category, sketch a triangle and write its angles, or explain why no example exists. Hint: think about why "right equilateral" and "obtuse equilateral" can't exist.

Reveal solution

Acute equilateral: $60, 60, 60$. Acute isosceles: e.g. $70, 70, 40$. Acute scalene: e.g. $50, 60, 70$. Right isosceles: $45, 45, 90$. Right scalene: e.g. $30, 60, 90$. Obtuse isosceles: e.g. $30, 30, 120$. Obtuse scalene: e.g. $20, 50, 110$. "Right equilateral" and "obtuse equilateral" don't exist — an equilateral has every angle $60^{\circ}$, none of which is $90^{\circ}$ or more.

Quick Review

Acute-angled

All three angles less than $90^{\circ}$.

Right-angled

One angle exactly $90^{\circ}$.

Obtuse-angled

One angle greater than $90^{\circ}$.

Hypotenuse

Side opposite the right angle — longest side.

At most ONE big angle

Never two right or two obtuse angles.

Combined name

e.g. "right isosceles", "obtuse scalene".

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