Mathematics • Year 7 • Unit 3 • Lesson 10

Angle Sum of Quadrilaterals

Build fluency with the rule ∠A + ∠B + ∠C + ∠D = 360°. Apply it to find a missing angle, set up algebraic angle equations, and confirm the proof (one diagonal splits a quadrilateral into two triangles, each 180°). Always state the reason: (∠ sum of quad).

Build · I Do / We Do / You Do

1. I do — fully worked example

Read every line. Each step uses the 360° rule with a reason in brackets.

Problem. A quadrilateral has angles 75°, 110°, 95° and x°. Find x.

Step 1 — State the rule.

∠A + ∠B + ∠C + ∠D = 360° (∠ sum of quad)

Reason: the four interior angles of every quadrilateral add to 360° (proof: one diagonal makes two triangles, 2 × 180° = 360°).

Step 2 — Substitute the known values.

75 + 110 + 95 + x = 360

Step 3 — Add the three known angles.

280 + x = 360

Step 4 — Solve.

x = 360 − 280 = 80°

Check: 75 + 110 + 95 + 80 = 360° ✓

Answer: x = 80°.

Stuck? Revisit lesson § "Find the fourth angle" — every quadrilateral, no matter what type, has angles summing to 360°.

2. We do — fill in the missing steps

Fill in each blank. This problem has angles given as expressions in x. 4 marks

Problem. A quadrilateral has angles (x + 30)°, (2x)°, (x + 50)° and (2x + 40)°. Find x and state each angle.

Step 1 — Apply the 360° rule.

(x + 30) + 2x + (x + 50) + (2x + 40) = _______ ° (∠ sum of quad)

Step 2 — Collect like terms (add the x coefficients, add the constants).

x coefficients: 1 + 2 + 1 + 2 = _______ | constants: 30 + 50 + 40 = _______

_______ x + _______ = 360

Step 3 — Move the constant.

_______ x = 360 − _______ = _______

Step 4 — Solve for x.

x = _______ ÷ _______ = _______

Step 5 — Substitute back for each angle.

Angles = (_______ + 30), 2 × _______, (_______ + 50), (2 × _______ + 40) = _______ °, _______ °, _______ °, _______ °

Check: _______ + _______ + _______ + _______ = 360° ✓

Stuck? Add the x coefficients separately from the constants. Then it's a normal linear equation.

3. You do — independent practice

Use the 360° rule with a reason in brackets after each line.

Foundation — find the fourth angle

3.1 Angles 60°, 70°, 80°, x°. Find x. 1 mark

3.2 Angles 90°, 90°, 90°, x°. Find x and name the shape. 1 mark

3.3 Angles x, x, x, x. Find x. 1 mark

3.4 State the angle sum rule for a quadrilateral and explain in one sentence WHY it works (by splitting the shape with a diagonal). 1 mark

Standard — set up an equation

3.5 Angles 2x°, 3x°, 4x°, 3x°. Find x and state each angle. 2 marks

3.6 Three angles of a quadrilateral are 95°, 95° and 85°. Find the fourth and explain whether the quadrilateral could be a parallelogram. 2 marks

Extension — algebra with constants

3.7 A kite has ∠A = 70°, ∠B = x°, ∠C = 110° and ∠D = x°. Find x. 2 marks

3.8 A quadrilateral has angles (x + 20)°, (2x − 10)°, (x + 40)° and (2x + 10)°. Find x and state each angle. 3 marks

Stuck on 3.8? Add x coefficients (1 + 2 + 1 + 2 = 6) and constants (20 − 10 + 40 + 10 = 60) separately.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do (x + 30) + 2x + (x + 50) + (2x + 40) = 360

Step 1: sum = 360° (∠ sum of quad).
Step 2: x coefficients sum = 6; constants sum = 120. So 6x + 120 = 360.
Step 3: 6x = 360 − 120 = 240.
Step 4: x = 240 ÷ 6 = 40.
Step 5: Angles = (40 + 30), 2 × 40, (40 + 50), (2 × 40 + 40) = 70°, 80°, 90°, 120°.
Check: 70 + 80 + 90 + 120 = 360° ✓

3.1 — 60° + 70° + 80° + x = 360°

x = 360 − 60 − 70 − 80 = 150° (∠ sum of quad).

3.2 — 90° + 90° + 90° + x = 360°

x = 360 − 270 = 90°. The shape is a rectangle (four right angles), and could be a square if the sides are also all equal.

3.3 — All four angles equal

4x = 360 (∠ sum of quad). x = 90°. Four equal angles → four right angles → at least a rectangle.

3.4 — State the rule and explain

The four interior angles of any quadrilateral sum to 360°. Reason: drawing one diagonal splits the quadrilateral into two triangles, each with angle sum 180°, so the total interior angle is 2 × 180° = 360°.

3.5 — 2x + 3x + 4x + 3x = 360

12x = 360, x = 30. Angles = 2(30), 3(30), 4(30), 3(30) = 60°, 90°, 120°, 90°. Check: 60 + 90 + 120 + 90 = 360° ✓.

3.6 — 95° + 95° + 85° + ? = 360°

Sum of three = 275. Fourth = 360 − 275 = 85°. For a parallelogram, opposite angles must be equal — here the four angles are 95°, 95°, 85°, 85°. If the two 95° angles are OPPOSITE each other (and the two 85° angles are opposite), then yes — could be a parallelogram. If two equal angles are adjacent instead, no.

3.7 — Kite: 70 + x + 110 + x = 360

2x + 180 = 360, so 2x = 180 and x = 90°. The kite has angles 70°, 90°, 110°, 90°. Check: 70 + 90 + 110 + 90 = 360° ✓.

3.8 — (x + 20) + (2x − 10) + (x + 40) + (2x + 10) = 360

Collect: 6x + 60 = 360 (∠ sum of quad). 6x = 300, x = 50.
Angles = (50 + 20), (100 − 10), (50 + 40), (100 + 10) = 70°, 90°, 90°, 110°. Check: 70 + 90 + 90 + 110 = 360° ✓.