Mathematics • Year 8 • Unit 3 • Lesson 15

Angles in Triangles

Build fluency with the angle sum (always 180°), the exterior angle theorem (= sum of two remote interior angles), and triangle classification (acute, right, obtuse; equilateral, isosceles, scalene).

Build · I Do / We Do / You Do

1. I do — fully worked example

Two big rules: (1) interior angles add to 180°; (2) an exterior angle = sum of the two REMOTE interior angles.

Problem. A triangle has angles 55°, 72°, and x°. Find x.

Step 1 — Write the angle-sum equation.

55 + 72 + x = 180

Reason: the sum of the interior angles of ANY triangle is 180°. Always.

Step 2 — Simplify.

127 + x = 180

Step 3 — Solve.

x = 180 − 127 = 53°

Step 4 — Classify the triangle.

All three angles (55°, 72°, 53°) are less than 90° → ACUTE. All three are different → SCALENE.

Answer: x = 53°; the triangle is acute and scalene.

Stuck? Revisit lesson § Card 4 — angle sum: missing angle = 180° − (sum of other two).

2. We do — fill in the missing steps

An isosceles triangle has a vertex angle of 80°. Find the two equal base angles. 4 marks

Step 1 — Write down what we know about isosceles triangles:

An isosceles triangle has ____ equal sides and ____ equal base angles.

Step 2 — Set up the angle-sum equation (let each base angle = b):

80 + b + b = ______ ⟶ 80 + 2b = ______

Step 3 — Solve for b:

2b = ______ − 80 = ______ ⟶ b = ______ ÷ 2 = ______°

Step 4 — State the answer:

Each base angle = ______°

Stuck? Isosceles base angles formula: each = (180° − vertex) / 2.

3. You do — independent practice

Show all working AND a brief reason. Foundation: angle sum. Standard: exterior and isosceles. Extension: algebra.

Foundation — find the missing angle

3.1 Triangle angles: 60°, 80°, x°. Find x. 1 mark

3.2 A triangle has two angles equal to 45° each. Find the third angle and classify the triangle. 1 mark

3.3 A right-angled triangle has one acute angle of 35°. Find the other acute angle. 1 mark

3.4 A triangle has angles 95°, 32°, and y°. Find y and classify the triangle. 2 marks

Standard — exterior angle and isosceles

3.5 An exterior angle of a triangle is 130°. One remote interior angle is 70°. Find the other remote interior angle. 2 marks

3.6 An isosceles triangle has a vertex angle of 50°. Find the two equal base angles. 2 marks

Extension — algebraic triangles

3.7 A triangle has angles x, 2x, and 3x. Find all three angles and classify the triangle. 2 marks

3.8 A triangle has angles (2x + 10)°, (x + 20)°, and x°. Find x, all angles, and classify by angle type. 2 marks

Stuck on 3.7 / 3.8? Sum the algebraic expressions and set equal to 180°. Solve for x, then substitute back.

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Answers — Do not peek before attempting

Section 2 — We do (isosceles, vertex 80°)

Step 1: 2 equal sides and 2 equal base angles.
Step 2: 80 + 2b = 180.
Step 3: 2b = 180 − 80 = 100; b = 100 ÷ 2 = 50°.
Step 4: Each base angle = 50°.

3.1 — Angles 60°, 80°, x°

x = 180° − 60° − 80° = 40°.

3.2 — Two angles = 45° each

Third angle = 180° − 45° − 45° = 90°. This is a right-angled isosceles triangle.

3.3 — Right triangle, one acute = 35°

Other acute angle = 180° − 90° − 35° = 55°. (Or: the two acute angles in a right triangle add to 90°, so other = 90° − 35° = 55°.)

3.4 — Angles 95°, 32°, y°

y = 180° − 95° − 32° = 53°. One angle (95°) is greater than 90°, so this is an obtuse scalene triangle.

3.5 — Exterior 130°, one remote = 70°

Exterior angle = sum of two remote interior angles: 130° = 70° + other → other = 60°.

3.6 — Isosceles, vertex 50°

Each base angle = (180° − 50°) / 2 = 130° / 2 = 65°.

3.7 — Angles x, 2x, 3x

x + 2x + 3x = 180 → 6x = 180 → x = 30°. Angles: 30°, 60°, 90°. One angle is 90°, all different → right-angled scalene triangle.

3.8 — Angles (2x+10), (x+20), x

(2x + 10) + (x + 20) + x = 180 → 4x + 30 = 180 → 4x = 150 → x = 37.5°. Angles: 2(37.5)+10 = 85°, 37.5+20 = 57.5°, 37.5°. All less than 90° → acute scalene.