Mathematics • Year 7 • Unit 3 • Lesson 5
Exterior Angles of Triangles
Build fluency with the Exterior Angle Theorem: an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. Use this directly, plus the supplementary relationship to the adjacent interior angle (their sum is 180°), to find unknowns quickly. State (ext. ∠ of △).
1. I do — fully worked example
Read every line. Each step ends with the reason (ext. ∠ of △) or (∠s on str. line).
Problem. A triangle has two interior angles 45° and 70°. One of the sides is extended past the third vertex to form an exterior angle. Find this exterior angle.
Step 1 — State the Exterior Angle Theorem.
Exterior angle = sum of the two REMOTE (non-adjacent) interior angles
Reason: ext. ∠ of △ rule.
Step 2 — Identify the two remote interiors.
The 45° and 70° are at the OTHER two vertices (not the one where the side was extended).
Step 3 — Add them.
45 + 70 = 115° (ext. ∠ of △)
Step 4 — Sanity check using the angle sum.
The adjacent interior would be 180 − 115 = 65°.
Check: 45 + 70 + 65 = 180° ✓
Answer: Exterior angle = 115°.
2. We do — fill in the missing steps
Faded working — fill the blanks. 4 marks
Problem. An exterior angle of a triangle is 130°. One of the remote interior angles is 55°. Find the other remote interior angle x.
Step 1 — Apply the Exterior Angle Theorem:
55 + x = _______ (ext. ∠ of △)
Step 2 — Solve for x:
x = 130 − ______ = ______°
Step 3 — Find the adjacent interior angle:
Adjacent interior = 180 − ______ = ______° (∠s on str. line)
Step 4 — Check using the angle sum:
55 + ______ + ______ = ______° ✓
3. You do — independent practice
Show working AND a reason for every step. Use (ext. ∠ of △) or (∠s on str. line) as needed.
Foundation — single application
3.1 Two remote interior angles of a triangle are 50° and 60°. Find the exterior angle. 1 mark
3.2 Two remote interior angles are 35° and 95°. Find the exterior angle. 1 mark
3.3 An exterior angle is 110°. What is the adjacent interior angle at the same vertex? 1 mark
3.4 An exterior angle is 145° and one remote interior is 80°. Find the other remote interior. 1 mark
Standard — two-step
3.5 A triangle has interior angles 30°, 70° and 80°. (a) Find the exterior angle at the vertex with the 80° interior. (b) Check using the supplementary relationship. 2 marks
3.6 An exterior angle of a triangle is 100°. One remote interior is 45°. (a) Find the other remote interior. (b) Find the adjacent interior at the same vertex as the exterior. 2 marks
Extension — algebra
3.7 An exterior angle of a triangle is 100°. The two remote interior angles are x° and (x + 20)°. Set up and solve an equation, then state both remote interiors. 3 marks
3.8 Explain in your own words why the Exterior Angle Theorem MUST be true, using the facts that (i) the angles of a triangle sum to 180°, and (ii) the exterior and adjacent interior angles sit on a straight line and sum to 180°. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — Exterior 130°, one remote 55°
Step 1: 55 + x = 130. Step 2: x = 130 − 55 = 75°.
Step 3: adjacent = 180 − 130 = 50° (∠s on str. line).
Step 4: 55 + 75 + 50 = 180° ✓.
3.1 — Remote 50°, 60°
Exterior = 50 + 60 = 110° (ext. ∠ of △).
3.2 — Remote 35°, 95°
Exterior = 35 + 95 = 130° (ext. ∠ of △).
3.3 — Exterior 110°, adjacent interior
Adjacent = 180 − 110 = 70° (∠s on str. line).
3.4 — Exterior 145°, one remote 80°
Other remote = 145 − 80 = 65° (ext. ∠ of △).
3.5 — Interior 30°, 70°, 80° — exterior at 80°
(a) Exterior at the 80° vertex = sum of the OTHER two interiors = 30 + 70 = 100° (ext. ∠ of △).
(b) Check: exterior + adjacent interior = 100 + 80 = 180° ✓ (∠s on str. line).
3.6 — Exterior 100°, one remote 45°
(a) Other remote = 100 − 45 = 55° (ext. ∠ of △).
(b) Adjacent interior = 180 − 100 = 80° (∠s on str. line). Check sum: 45 + 55 + 80 = 180° ✓.
3.7 — Exterior 100°, remotes x°, (x + 20)°
x + (x + 20) = 100 (ext. ∠ of △).
Combine: 2x + 20 = 100 → 2x = 80 → x = 40.
Remote interiors: 40° and 40 + 20 = 60°. Adjacent interior = 180 − 100 = 80°. Check 40 + 60 + 80 = 180 ✓.
3.8 — Why the Exterior Angle Theorem must be true
Let the three interior angles of the triangle be a, b and c, where c is the interior angle at the vertex that has the exterior angle. Then by the angle sum, a + b + c = 180°. By the straight-line relationship at that vertex, exterior + c = 180°. Subtracting the second equation from the first gives: a + b = exterior. So the exterior angle equals the sum of the two remote interior angles (a and b) — the ones at the OTHER two vertices, not at vertex c. That's the Exterior Angle Theorem.