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.