Mathematics • Year 7 • Unit 3 • Lesson 4

The Angle Sum of Triangles

Build fluency with the rule: the three interior angles of any triangle add to 180°. Find unknown angles by subtraction, find equal base angles of an isosceles triangle, and solve algebraic angle problems like x + 2x + 30 = 180. Always state (angle sum of △).

Build · I Do / We Do / You Do

1. I do — fully worked example

Read every line — each step uses the 180° rule and ends with a clearly stated reason.

Problem. Two angles of a triangle are 48° and 76°. Find the third angle.

Step 1 — State the rule.

Sum of interior angles of any triangle = 180°

Reason: angle sum of △.

Step 2 — Add the two known angles.

48 + 76 = 124°

Step 3 — Subtract from 180°.

180 − 124 = 56° (angle sum of △)

Check: 48 + 76 + 56 = 180° ✓.

Answer: The third angle is 56°.

Stuck? Revisit lesson § "Watch Me Solve It · Find the third angle" — always state (angle sum of △) next to the calculation.

2. We do — fill in the missing steps

Faded working — fill the blanks. 4 marks

Problem. An isosceles triangle has an apex angle of 34°. Find the two base angles.

Step 1 — Let each base angle be B and set up the equation:

34 + B + B = _______ (angle sum of △)

34 + 2B = _______

Step 2 — Solve for 2B:

2B = 180 − ______ = ______

Step 3 — Solve for B:

B = ______ ÷ 2 = ______°

Step 4 — Check: 34 + ______ + ______ = 180° ✓

Stuck? Revisit lesson § "Watch Me Solve It · Isosceles" — the two base angles are equal, so use 2B for them both.

3. You do — independent practice

Always state (angle sum of △) next to your final calculation. Show your working.

Foundation — single subtraction

3.1 Two angles of a triangle are 70° and 60°. Find the third. 1 mark

3.2 Two angles of a triangle are 90° and 45°. Find the third. 1 mark

3.3 Two angles of a triangle are 110° and 25°. Find the third. 1 mark

3.4 An equilateral triangle has all three angles equal. Find the size of each. 1 mark

Standard — isosceles and unknowns

3.5 An isosceles triangle has two base angles of 65° each. Find the apex angle. 2 marks

3.6 An isosceles triangle has an apex angle of 80°. Find each base angle. 2 marks

Extension — algebra

3.7 The angles of a triangle are x°, 2x° and 30°. Set up an equation using the angle sum, solve for x, and state all three angles. 3 marks

3.8 Could a triangle have angles 70°, 80° and 40°? Justify with the angle sum. If not, what is wrong? 2 marks

Stuck on 3.8? Add them up. Does the total equal 180°?

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Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — Isosceles apex 34°

Step 1: 34 + 2B = 180. Step 2: 2B = 180 − 34 = 146. Step 3: B = 146 ÷ 2 = 73°. Step 4: 34 + 73 + 73 = 180° ✓.

3.1 — 70°, 60°, ?

180 − 70 − 60 = 50° (angle sum of △).

3.2 — 90°, 45°, ?

180 − 90 − 45 = 45° (angle sum of △). (Right isosceles.)

3.3 — 110°, 25°, ?

180 − 110 − 25 = 45° (angle sum of △).

3.4 — Equilateral, all angles equal

Each angle = 180 ÷ 3 = 60° (angle sum of △).

3.5 — Base angles 65° each, find apex

Apex = 180 − 65 − 65 = 50° (angle sum of △).

3.6 — Apex 80°, find base angles

Let each base angle = B. Then 80 + 2B = 180 → 2B = 100 → B = 50°. So each base angle is 50° (angle sum of △).

3.7 — Angles x°, 2x°, 30°

x + 2x + 30 = 180 (angle sum of △). Combine: 3x + 30 = 180 → 3x = 150 → x = 50. Angles: 50°, 100°, 30°. Check: 50 + 100 + 30 = 180° ✓.

3.8 — 70°, 80°, 40°?

70 + 80 + 40 = 190°, which is more than 180°. Since the angle sum of any triangle must be exactly 180°, this set of angles cannot form a triangle.