Mathematics • Year 7 • Unit 3 • Lesson 3
Types of Triangles by Their Angles
Build fluency classifying triangles as acute-angled (all angles < 90°), right-angled (exactly one angle = 90°), or obtuse-angled (one angle > 90°). Identify the hypotenuse as the side opposite the right angle, and combine angle-classification with side-classification (e.g. "right isosceles").
1. I do — fully worked example
Read every line. Each step compares an angle to 90° to decide the classification.
Problem. A triangle has interior angles 40°, 60° and 80°. Classify it by its angles, and explain whether it could also be isosceles.
Step 1 — Check the angle sum is 180°.
40 + 60 + 80 = 180° ✓
Reason: a valid triangle must add to 180°.
Step 2 — Test each angle against 90°.
40 < 90, 60 < 90, 80 < 90 → all three less than 90°
Step 3 — Name the type.
All angles < 90° → acute-angled
Step 4 — Could it be isosceles?
All three angles are different (40, 60, 80) → no two equal → not isosceles → scalene by sides.
Reason: equal angles would force equal opposite sides.
Answer: acute-angled scalene.
2. We do — fill in the missing steps
Same structure, faded working. Fill the blanks. 4 marks
Problem. A right-angled triangle has legs 6 cm and 8 cm with the right angle between them, and a third side of 10 cm. Identify the hypotenuse and explain why it must be the longest side.
Step 1 — Locate the right angle: the right angle is between the ____ cm and ____ cm sides.
Step 2 — Look across (opposite) the right angle:
The side opposite the right angle is ____ cm → this is the ___________________.
Step 3 — Check it is the longest:
____ > ____ > ____ ✓
Step 4 — Write the rule: the hypotenuse is always opposite the ___________________ and is the ___________________ side.
3. You do — independent practice
Show your working. Always test each angle against 90° before naming the type.
Foundation — single comparison
3.1 Classify by angles: 50°, 60°, 70°. 1 mark
3.2 Classify by angles: 90°, 45°, 45°. 1 mark
3.3 Classify by angles: 110°, 40°, 30°. 1 mark
3.4 A right-angled triangle has sides 9 cm, 12 cm and 15 cm (right angle between the 9 and 12). Which side is the hypotenuse? 1 mark
Standard — combine with sides
3.5 A triangle has angles 30°, 30° and 120°. Classify it by BOTH angles AND sides. 2 marks
3.6 A triangle has angles 60°, 60° and 60°. Classify it by BOTH angles AND sides. 2 marks
Extension — push your thinking
3.7 Is it possible for a triangle to have TWO right angles? Justify your answer using the angle sum of 180°. 2 marks
3.8 A right isosceles triangle has its right angle marked with a small square. Without using the angle sum directly, explain why the two non-right angles must be 45° each. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — Hypotenuse of 6-8-10
Step 1: between the 6 cm and 8 cm sides.
Step 2: side opposite = 10 cm → hypotenuse.
Step 3: 10 > 8 > 6 ✓.
Step 4: opposite the right angle; always the longest side.
3.1 — 50, 60, 70
All angles < 90° → acute-angled.
3.2 — 90, 45, 45
One angle = 90° → right-angled.
3.3 — 110, 40, 30
One angle > 90° → obtuse-angled.
3.4 — Hypotenuse of 9-12-15
The right angle is between the 9 and 12, so the side opposite is the 15 cm side — this is the hypotenuse (also the longest side).
3.5 — Angles 30, 30, 120
One angle (120°) > 90° → obtuse-angled. Two equal angles (30°, 30°) → equal opposite sides → isosceles. Combined: obtuse isosceles.
3.6 — Angles 60, 60, 60
All angles < 90° → acute-angled. All angles equal → all sides equal → equilateral. Combined: acute equilateral (the equilateral triangle is always acute).
3.7 — Two right angles?
No. Two right angles would already total 90 + 90 = 180°, leaving 0° for the third angle. A triangle must have three angles greater than 0°, so two right angles is impossible.
3.8 — Why right isosceles is 45-45-90
The two equal sides of the isosceles triangle force the two angles opposite them to be equal (base angles of an isosceles triangle are equal). Those two equal angles plus the 90° right angle must total 180°, so the two equal angles together total 180 − 90 = 90°. Splitting 90° equally between them gives 90 ÷ 2 = 45° each.