Mathematics • Year 7 • Unit 3 • Lesson 3

Triangles by Angles — Mixed Challenge

Bring together angle classification, hypotenuse identification, and combined labels. Spot a common 90° error and tackle an open-ended classification puzzle.

Master · Mixed Challenge

1. Mixed problems — choose the right rule

Decide angle type, side type, and hypotenuse where relevant. 2 marks each

1.1 Classify by angles: 25°, 65°, 90°.

1.2 Classify by angles: 50°, 65°, 65°. Then classify by sides.

1.3 A triangle has angles 130°, 30°, 20°. Combine angle and side classification.

1.4 A right-angled triangle has sides 8 cm, 15 cm and 17 cm (right angle between the 8 and 15). Identify the hypotenuse and justify in one sentence.

1.5 Two angles of a triangle are 90° and 45°. Find the third angle and classify the triangle in two ways.

1.6 Explain why no triangle can have an angle that is exactly 180°.

Stuck on 1.6? 180° would use the entire angle sum on its own, leaving 0° for the other two angles.

2. Find the mistake

Another Year 7 student tried to classify a triangle. Exactly one line contains a mistake. Spot it, explain why, then re-do the working correctly. 3 marks

Student's problem: A triangle has angles 90°, 45°, 45°. Classify it by angles AND sides, and identify the hypotenuse.

Line 1: One angle is 90° → right-angled.

Line 2: All three angles are different (90, 45, 45) → scalene.

Line 3: Hypotenuse is opposite the 90° angle and is the longest side.

(a) Which line contains the mistake?

(b) Explain in one or two sentences why that line is wrong.

(c) Write out the corrected working in full, including the corrected final label and the hypotenuse identification.

Stuck? Re-count the equal angles in 90, 45, 45. How many are equal?

3. Open-ended challenge — find all triangle types

This question has many correct answers. 4 marks

3.1 Construct one set of valid interior angles (positive whole numbers summing to 180°) for EACH of the six possible combined triangle types listed below. Write the angles and verify they sum to 180°.

(i) Acute scalene
(ii) Acute isosceles
(iii) Acute equilateral
(iv) Right scalene
(v) Right isosceles
(vi) Obtuse isosceles

Bonus: Is "obtuse equilateral" possible? Justify your answer in one sentence.

Stuck? Equal angles ↔ equal sides. Right means one 90°. Obtuse means one > 90°. Equilateral forces all three angles to 60°.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — 25, 65, 90

One angle = 90° → right-angled.

1.2 — 50, 65, 65

All angles < 90° → acute-angled. Two equal angles → isosceles. Combined: acute isosceles.

1.3 — 130, 30, 20

One angle (130°) > 90° → obtuse-angled. All three different → scalene. Combined: obtuse scalene.

1.4 — Hypotenuse of 8-15-17

The right angle sits between the 8 and the 15, so the side opposite it is the 17 cm side — this is the hypotenuse (also the longest).

1.5 — Angles 90, 45, ?

Third angle = 180 − 90 − 45 = 45°. By angles: right-angled. By sides: two equal angles → isosceles. Combined: right isosceles.

1.6 — Why no 180° angle

The three interior angles must sum to 180°. If one angle were 180° on its own, the other two would have to sum to 0°. That would require two angles of 0°, which means two "straight rays" with no rotation between them at the vertex — no triangle can form.

2 — Find the mistake

(a) The mistake is on Line 2.
(b) The student claimed "all three angles are different" but in fact 45° = 45° (two equal angles). Two equal angles force two equal opposite sides, so the triangle is isosceles, not scalene.
(c) Corrected working:
Line 1: One angle is 90° → right-angled.
Line 2: Two equal angles (45°, 45°) → equal opposite sides → isosceles. Combined label: right isosceles.
Line 3: Hypotenuse is opposite the 90° and is the longest side (the two 45° angles sit opposite the two equal legs).

3 — Open-ended challenge (sample answers)

Many valid answers; here is one valid set per type:
(i) Acute scalene: 50°, 60°, 70° (sum 180 ✓; all < 90; all different).
(ii) Acute isosceles: 70°, 70°, 40° (sum 180 ✓; all < 90; two equal).
(iii) Acute equilateral: 60°, 60°, 60° (sum 180 ✓; all < 90; all equal).
(iv) Right scalene: 30°, 60°, 90° (sum 180 ✓; one 90°; all different).
(v) Right isosceles: 45°, 45°, 90° (sum 180 ✓; one 90°; two equal).
(vi) Obtuse isosceles: 20°, 20°, 140° (sum 180 ✓; one > 90; two equal).
Bonus: "Obtuse equilateral" is not possible. An equilateral triangle has all three angles equal, and three equal angles summing to 180° must each be 60° — which is acute, not obtuse.

Marking: 1 mark each for any two correct sets across (i)–(vi), capped at 3; 1 mark for the correct bonus justification.