Mathematics • Year 7 • Unit 3 • Lesson 5
Exterior Angles — Mixed Challenge
Bring together the Exterior Angle Theorem, the 180° straight-line relationship, the 180° angle sum of a triangle, and algebraic angle expressions. Spot a vertex-confusion error and tackle an open-ended construction puzzle.
1. Mixed problems — choose the right rule
Decide which to use: (ext. ∠ of △), (∠s on str. line), or (angle sum of △). Show working. 2 marks each
1.1 Two remote interior angles of a triangle are 35° and 75°. Find the exterior angle.
1.2 An exterior angle is 145°. Find the adjacent interior angle.
1.3 An exterior angle is 95° and one remote interior is 40°. Find the other remote and the adjacent interior.
1.4 An exterior angle of a triangle is 110°. The two remote interior angles are equal. Find each.
1.5 The exterior angle of a triangle is (3x + 10)° and the two remote interiors are 50° and 80°. Find x and state the exterior angle.
1.6 An exterior angle of a triangle is 70°. Is the adjacent interior angle acute, right, or obtuse? Justify.
2. Find the mistake
Another Year 7 student has tried to apply the Exterior Angle Theorem. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why, then re-do the working correctly. 3 marks
Student's problem: A triangle has interior angles 40°, 60° and 80°. The 40° vertex has its side extended, forming an exterior angle. Find the exterior angle at the 40° vertex.
Line 1: By the Exterior Angle Theorem, exterior = sum of remote interior angles.
Line 2: Remote interiors are the ones NOT at the same vertex as the exterior.
Line 3: So the remote interiors are 40° and 60° (the two smaller ones).
Line 4: Exterior at 40° vertex = 40 + 60 = 100°.
(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 exterior angle.
Stuck? The exterior is at the 40° vertex — so 40° is the ADJACENT interior, not a remote interior. The remotes are the OTHER two.3. Open-ended challenge — design a triangle from one exterior angle
This question has many correct answers. 4 marks
3.1 A triangle has one of its sides extended past a vertex, creating an exterior angle of exactly 120°. Find three different valid sets of interior angles for the triangle that are consistent with this. For each set:
- (i) state the three interior angles (positive whole numbers, summing to 180°);
- (ii) verify the Exterior Angle Theorem holds: the two remote interiors must sum to 120°;
- (iii) verify the adjacent interior (the third angle) is 180 − 120 = 60°.
Bonus: Of your three valid triangles, are any of them isosceles by angles? Identify any that are.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Remote 35°, 75°
Exterior = 35 + 75 = 110° (ext. ∠ of △).
1.2 — Exterior 145°
Adjacent interior = 180 − 145 = 35° (∠s on str. line).
1.3 — Exterior 95°, one remote 40°
Other remote = 95 − 40 = 55° (ext. ∠ of △). Adjacent interior = 180 − 95 = 85° (∠s on str. line). Check: 40 + 55 + 85 = 180° ✓.
1.4 — Exterior 110°, equal remotes
Let each remote = R. 2R = 110 → R = 55°. So each remote interior is 55° (ext. ∠ of △).
1.5 — Exterior (3x + 10)°, remotes 50° and 80°
3x + 10 = 50 + 80 = 130 (ext. ∠ of △). So 3x = 120 → x = 40. Exterior = 3(40) + 10 = 130°.
1.6 — Exterior 70°, adjacent interior
Adjacent = 180 − 70 = 110°. Since 110° > 90°, the adjacent interior is obtuse.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The 40° is at the SAME vertex as the exterior angle, so 40° is the adjacent interior, not a remote interior. The two remote interiors are the angles at the OTHER two vertices — namely 60° and 80°.
(c) Corrected working:
Line 1: Exterior = sum of remote interiors (ext. ∠ of △).
Line 2: Remote interiors are at the OTHER two vertices, NOT at the vertex with the exterior.
Line 3 (fixed): Remote interiors are 60° and 80° (the two NOT at the 40° vertex).
Line 4 (fixed): Exterior at 40° vertex = 60 + 80 = 140°. Check: 40 + 140 = 180° ✓ (∠s on str. line).
3 — Open-ended challenge (sample answers)
Adjacent interior is fixed at 180 − 120 = 60°. The other two angles must sum to 120° (and together with the 60° give 180° total). Three sample valid sets:
Set 1: 60°, 60°, 60° (equilateral). Remote pair sums to 60 + 60 = 120 ✓; adjacent = 60 ✓.
Set 2: 60°, 50°, 70°. Remotes (50, 70) sum to 120 ✓; adjacent = 60 ✓.
Set 3: 60°, 40°, 80°. Remotes (40, 80) sum to 120 ✓; adjacent = 60 ✓.
Bonus: Set 1 (60°, 60°, 60°) is equilateral — and equilateral is a special case of isosceles (it has more than two equal angles). If we accept only strictly-two-equal-angle isosceles, then Set 1 alone is isosceles. Another valid isosceles set would be 60°, 60°, 60° (already listed) or 60°, 60°, 60°. A new strictly-isosceles option: 60°, 80°, 40° is scalene; try 60°, 70°, 50° — also scalene; try the "two equal remotes" case: 60°, 60°, 60° again. So only Set 1 in our list is isosceles. (Students who supply 60°, 30°, 90° as a fourth set with a right angle should get credit; 60°, 30°, 90°: remotes 30 + 90 = 120 ✓, adjacent 60 ✓.)
Marking: 1 mark for each genuinely different valid set (3 marks total); 1 mark for the bonus identification with a clear yes/no justification.