Mathematics • Year 8 • Unit 3 • Lesson 16
Angles in Quadrilaterals
Build fluency with the rule "interior angles of any quadrilateral add to 360°". One fully worked example, one guided example with blanks, then eight independent problems ramping from clean number questions to algebraic angle hunts.
1. I do — fully worked example
Read every line. Each step has a short reason so you can see why we do it, not just what.
Problem. A quadrilateral has three known angles: 110°, 85°, and 70°. Find the fourth angle x.
Step 1 — Write the angle-sum rule.
Sum of interior angles of a quadrilateral = 360°
Reason: every quadrilateral splits into 2 triangles, so 2 × 180° = 360°.
Step 2 — Set up the equation.
110 + 85 + 70 + x = 360
Reason: the four interior angles must add to exactly 360°.
Step 3 — Add the known angles.
265 + x = 360
Reason: combine the numbers you know to simplify the equation.
Step 4 — Subtract from 360°.
x = 360 − 265 = 95°
Reason: undo the addition to isolate x. Always check: 110 + 85 + 70 + 95 = 360. ✓
Answer: x = 95°
2. We do — fill in the missing steps
Same shape as Section 1, with the working faded. Fill in each blank. 4 marks
Problem. A quadrilateral has three known angles: 120°, 95°, and 60°. Find the fourth angle x.
Step 1 — Rule: interior angles of any quadrilateral add to ______°.
Step 2 — Set up the equation:
120 + 95 + 60 + x = ______
Step 3 — Add the known angles:
______ + x = 360
Step 4 — Subtract:
x = 360 − ______ = ______°
3. You do — independent practice
Show all working. The first three are foundation (one missing angle). The middle three are standard (special quadrilaterals and equal-angle setups). The last two are extension (algebra with x).
Foundation — find the missing angle
3.1 Three angles of a quadrilateral are 100°, 80°, and 90°. Find the fourth angle. 1 mark
3.2 Three angles of a quadrilateral are 130°, 70°, and 55°. Find the fourth angle. 1 mark
3.3 A quadrilateral has angles 145°, 75°, and 65°. Find the fourth angle. 1 mark
Standard — special quadrilaterals and equal angles
3.4 A parallelogram has one angle of 65°. State the sizes of the other three angles. (Hint: opposite angles equal; co-interior angles add to 180°.) 2 marks
3.5 Quadrilateral PQRS has ∠P = 105° and ∠R = 80°. If ∠Q = ∠S, find the size of ∠Q. 2 marks
3.6 A kite has angles 110°, 75°, 110° and x°. Find x. (Recall: in a kite, one pair of opposite angles is equal.) 2 marks
Extension — algebraic angles
3.7 A quadrilateral has angles x°, 2x°, 90° and 90°. Find x. 2 marks
3.8 A quadrilateral has angles (2x + 10)°, 3x°, (x + 20)° and 90°. Find x and list all four angles. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (120°, 95°, 60°, x)
Step 1: angles add to 360°.
Step 2: 120 + 95 + 60 + x = 360.
Step 3: 275 + x = 360.
Step 4: x = 360 − 275 = 85°.
3.1 — 100°, 80°, 90°
x = 360 − (100 + 80 + 90) = 360 − 270 = 90°.
3.2 — 130°, 70°, 55°
x = 360 − (130 + 70 + 55) = 360 − 255 = 105°.
3.3 — 145°, 75°, 65°
x = 360 − (145 + 75 + 65) = 360 − 285 = 75°.
3.4 — Parallelogram with one 65° angle
Opposite angle = 65°. Co-interior angles are supplementary: 180 − 65 = 115°, and its opposite is also 115°. Four angles: 65°, 115°, 65°, 115°. Check: 65 + 115 + 65 + 115 = 360 ✓
3.5 — PQRS with ∠P = 105°, ∠R = 80°, ∠Q = ∠S
105 + ∠Q + 80 + ∠Q = 360, so 2∠Q = 360 − 185 = 175, and ∠Q = ∠S = 87.5°.
3.6 — Kite with 110°, 75°, 110°, x°
x = 360 − (110 + 75 + 110) = 360 − 295 = 65°.
3.7 — Angles x°, 2x°, 90°, 90°
x + 2x + 90 + 90 = 360, so 3x + 180 = 360, 3x = 180, x = 60°. Angles: 60°, 120°, 90°, 90°.
3.8 — (2x+10)°, 3x°, (x+20)°, 90°
(2x+10) + 3x + (x+20) + 90 = 360, so 6x + 120 = 360, 6x = 240, x = 40. Angles: 2(40)+10 = 90°, 3(40) = 120°, 40+20 = 60°, 90°. Check: 90 + 120 + 60 + 90 = 360 ✓