Mathematics • Year 7 • Unit 3 • Lesson 5

Exterior Angles — Real World

Apply the Exterior Angle Theorem to real situations — roof eaves, traffic-signal supports, surveying, garden-bed extensions, and ramp overhangs. Use both forms: (i) "sum of remote interiors = exterior" and (ii) "exterior + adjacent interior = 180°".

Apply · Real-World Maths

1. Word problems

Identify the exterior angle, the adjacent interior, and the two remote interiors. Show working and reasons.

1.1 — Roof eave overhang. A roof truss is a triangle. The two roof angles inside the truss (at the wall corners) measure 60° and 75°. The wood is extended past the third corner (the peak) to form an eave, creating an exterior angle. Find the exterior eave angle. 2 marks

Stuck? The 60° and 75° are the two remote interiors (away from the peak).

1.2 — Traffic-signal support. A triangular metal support holds a traffic signal in place. The triangle has interior angles 50°, 50° and 80°. One of the sides at the 80° vertex is extended into a horizontal mounting arm. (a) Find the exterior angle at that vertex. (b) Confirm using the supplementary relationship. 2 marks

Stuck? At the 80° vertex, the remotes are the two 50°s.

1.3 — Surveyor's extension. A surveyor sets up a triangle between three points. One side is extended past a vertex to mark a fourth landmark, making an exterior angle of 120°. The interior angle at the OTHER end (one of the remote interiors) is 45°. Find the other remote interior angle. 2 marks

Stuck? remote₁ + remote₂ = exterior. So remote₂ = 120 − 45.

1.4 — Garden bed. A triangular garden bed has interior angles 65°, 55° and 60°. The landscaper extends each side outward to mark out a path going around the bed. Find the three exterior angles, one at each vertex, and verify their sum. 3 marks

Stuck? At each vertex, exterior = 180 − interior. Find all three, then add — the sum should be 360°.

1.5 — Skateboard ramp side panel. The side panel of a ramp is a right-angled triangle with interior angles 90°, 20° and 70°. The base of the ramp (the side meeting the ground) is extended forward to mark out where another panel will be welded. The extension forms an exterior angle at the 70° vertex. (a) Find this exterior angle. (b) Identify the two remote interiors. (c) Verify using the angle sum. 3 marks

Stuck on (b)? The two remote interiors are the ones at the OTHER two vertices: 90° and 20°.

2. Explain your thinking

Use full sentences. 4 marks

2.1 A classmate is asked to find an exterior angle of a triangle whose interior angles are 60°, 70° and 50°. They write "Exterior angle = 60 + 70 + 50 = 180°." Explain in your own words (i) what is wrong about adding all three interior angles, (ii) which angles the Exterior Angle Theorem actually involves (use the word remote), (iii) which TWO of the three interiors are the remote interiors for the exterior angle at the 60° vertex, and (iv) what the correct exterior angle at the 60° vertex really is.

Stuck? Three interior angles always sum to 180° — that's a different rule. Exterior = sum of just the TWO remote interiors.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Roof eave

Exterior at the peak = 60 + 75 = 135° (ext. ∠ of △).

1.2 — Traffic-signal support 50°, 50°, 80°

(a) Exterior at the 80° vertex = 50 + 50 = 100° (ext. ∠ of △).
(b) Adjacent interior + exterior = 80 + 100 = 180° ✓ (∠s on str. line).

1.3 — Surveyor's extension

Other remote = 120 − 45 = 75° (ext. ∠ of △).

1.4 — Garden bed 65°, 55°, 60°

Exterior at 65° vertex = 180 − 65 = 115°.
Exterior at 55° vertex = 180 − 55 = 125°.
Exterior at 60° vertex = 180 − 60 = 120°.
Sum of exterior angles = 115 + 125 + 120 = 360° ✓ (the three exterior angles of any triangle always sum to 360°).

1.5 — Ramp panel 90°, 20°, 70°

(a) Exterior at 70° vertex = 90 + 20 = 110° (ext. ∠ of △).
(b) The two remote interiors are 90° and 20°.
(c) Verify: adjacent interior + exterior = 70 + 110 = 180° ✓; full angle sum 90 + 20 + 70 = 180° ✓.

2.1 — Explain your thinking (sample response)

The classmate added all three interior angles — but that always equals 180° for any triangle (the angle sum rule), which doesn't tell you anything about a specific exterior angle. The Exterior Angle Theorem says an exterior angle equals the sum of just the TWO remote interior angles — the ones at the OTHER two vertices, not at the vertex where the exterior angle sits. For the exterior angle at the 60° vertex, the two remote interiors are the 70° and the 50° (the ones at the other two corners), so the correct exterior angle = 70 + 50 = 120°. Check: 60 + 120 = 180° ✓ on the straight line.

Marking: 1 for naming "angle sum" as the rule they wrongly used; 1 for using "remote"; 1 for correctly identifying 70° and 50° as the remotes for the 60° vertex; 1 for the correct exterior 120° with verification.