Mathematics • Year 8 • Unit 3 • Lesson 15
Angles in Triangles — Mixed Challenge
Six mixed problems across angle sum, exterior angle theorem, isosceles, equilateral and algebraic triangles. One "find the mistake" and one open-ended impossible-triangle hunt.
1. Mixed problems — choose the right move
Decide which theorem applies (angle sum, exterior angle, isosceles, equilateral) BEFORE you start. Show working. 3 marks each
1.1 A triangle has angles 38°, 65°, and y°. Find y and classify the triangle.
1.2 An exterior angle of a triangle is 145°. One of the remote interior angles is 80°. Find the other remote interior angle and the adjacent interior angle.
1.3 An isosceles triangle has a vertex angle of 100°. Find the two equal base angles.
1.4 A triangle has angles in the ratio 1 : 2 : 3. Find all three angles, and identify the triangle's type by angles.
1.5 An isosceles triangle has one BASE angle of 25°. Find the other base angle and the vertex angle. (Hint: in an isosceles triangle the TWO BASE ANGLES are equal — so the other base angle is also 25°.)
1.6 A triangle has angles (3x + 5)°, (2x + 25)°, and 70°. Find x and all three angles. Confirm the angle sum.
2. Find the mistake
A Year 8 student tried to find the base angles of an isosceles triangle with vertex angle 40°. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, and re-do the working. 3 marks
Student's working — isosceles triangle, vertex angle = 40°, find base angles:
Line 1: Sum of angles in a triangle = 180°
Line 2: Base angles + 40 = 180
Line 3: Base angles = 180 − 40 = 140°
Line 4: So each base angle is 140°.
(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 correct base-angle value.
Stuck? The student forgot there are TWO equal base angles. The "Base angles" total in Line 2 should be the sum of both base angles (2b), not just one b.3. Open-ended challenge — possible or impossible triangles?
This question has more than one valid answer. 4 marks
3.1 For each of the following angle sets, determine whether a triangle with those interior angles is POSSIBLE or IMPOSSIBLE. If possible, classify the triangle (acute / right / obtuse AND equilateral / isosceles / scalene). If impossible, explain why in one sentence.
(a) 60°, 60°, 60°
(b) 90°, 50°, 50°
(c) 100°, 40°, 40°
(d) 30°, 60°, 90°
(e) 70°, 70°, 60°
(f) 120°, 30°, 40°
Bonus: create your OWN angle set that is impossible, AND explain in one sentence which rule it breaks.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Angles 38°, 65°, y°
y = 180 − 38 − 65 = 77°. All angles < 90° → acute scalene.
1.2 — Exterior 145°, one remote = 80°
Other remote = 145° − 80° = 65°. Adjacent interior = 180° − 145° = 35°. Check: 80 + 65 + 35 = 180 ✓.
1.3 — Isosceles, vertex 100°
Each base angle = (180 − 100) / 2 = 80 / 2 = 40°.
1.4 — Ratio 1 : 2 : 3
x + 2x + 3x = 6x = 180 → x = 30°. Angles: 30°, 60°, 90°. One = 90° → right-angled (scalene since all different).
1.5 — Isosceles, one base = 25°
Other base = 25° (two base angles are equal). Vertex = 180° − 25° − 25° = 130°.
1.6 — (3x+5)°, (2x+25)°, 70°
5x + 100 = 180 → 5x = 80 → x = 16. Angles: 3(16)+5 = 53°, 2(16)+25 = 57°, 70°. Check: 53 + 57 + 70 = 180 ✓.
2 — Find the mistake
(a) The mistake is on Line 2 (carried into Lines 3 and 4).
(b) The student wrote "Base angles + 40 = 180" treating "base angles" as a single value. But isosceles triangles have TWO equal base angles, so the equation should be 2b + 40 = 180 (where b is one base angle).
(c) Corrected: 2b + 40 = 180 → 2b = 140 → b = 70°. Each base angle = 70°.
3 — Possible vs impossible triangles
(a) 60° + 60° + 60° = 180° ✓ → possible; acute equilateral.
(b) 90° + 50° + 50° = 190° ✗ → impossible (sum is 190°, not 180°).
(c) 100° + 40° + 40° = 180° ✓ → possible; obtuse isosceles.
(d) 30° + 60° + 90° = 180° ✓ → possible; right scalene.
(e) 70° + 70° + 60° = 200° ✗ → impossible (sum is 200°).
(f) 120° + 30° + 40° = 190° ✗ → impossible (sum is 190°).
Bonus example: 90°, 90°, 90° is impossible — sum is 270°, AND a triangle cannot have two right angles (the remaining angle would be 0°, which collapses the triangle to a line).
Marking: 0.5 mark each for parts (a)–(f) classified correctly (3 marks). 1 mark for valid bonus impossible triangle with reason.