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How many degrees do the angles of a triangle add to? What about a quadrilateral — can you guess what their angle sum might be?
Any four-sided shape (quadrilateral) can be split into exactly two triangles. Since each triangle's angles add to 180°, the interior angle sum of every quadrilateral is $2 \times 180° = 360°$. This is true whether the quadrilateral is a square, a kite, or a wonky irregular blob.
Many students think irregular quadrilaterals have a different angle sum from regular ones. They don't! Every quadrilateral — no matter how lopsided — has interior angles summing to exactly 360°. The proof works because you can always draw a diagonal to make two triangles, regardless of the shape.
Draw any quadrilateral $ABCD$. Draw the diagonal $AC$ to split it into two triangles: $\triangle ABC$ and $\triangle ACD$.
In $\triangle ABC$: $\angle BAC + \angle ABC + \angle BCA = 180°$
In $\triangle ACD$: $\angle CAD + \angle ACD + \angle ADC = 180°$
Adding both equations:
$$(\angle BAC + \angle CAD) + \angle ABC + (\angle BCA + \angle ACD) + \angle ADC = 360°$$But $\angle BAC + \angle CAD = \angle DAB$ and $\angle BCA + \angle ACD = \angle BCD$, so:
$$\angle DAB + \angle ABC + \angle BCD + \angle ADC = 360°$$This proves the interior angle sum of any quadrilateral is 360°.
| Shape | Angle Properties |
|---|---|
| Square | All four angles are 90° (4 × 90° = 360°) |
| Rectangle | All four angles are 90° |
| Rhombus | Opposite angles are equal; adjacent angles are supplementary (add to 180°) |
| Parallelogram | Opposite angles equal; co-interior angles (same-side) add to 180° |
| Trapezium | One pair of parallel sides; co-interior angles between parallel sides add to 180° |
| Kite | One pair of opposite angles are equal; the other two are different |
A quadrilateral has three known angles: 110°, 85°, and 70°. Find the fourth angle $x$.
Step 1: Use the angle sum rule.
$$110° + 85° + 70° + x = 360°$$Step 2: Add the known angles.
$$265° + x = 360°$$Step 3: Subtract from 360°.
$$x = 360° - 265° = \mathbf{95°}$$Always check: $110 + 85 + 70 + 95 = 360$ ✓
A quadrilateral has angles $(2x+10)°$, $3x°$, $(x+20)°$, and $90°$. Find $x$ and all four angles.
Step 1: Set up the equation.
$$(2x+10) + 3x + (x+20) + 90 = 360$$Step 2: Collect like terms.
$$6x + 120 = 360$$Step 3: Solve.
$$6x = 240 \implies x = 40$$Step 4: Substitute back.
Check: $90 + 120 + 60 + 90 = 360°$ ✓
Angle sum of any quadrilateral = 360°
Proof: Split into 2 triangles using a diagonal → $2 \times 180° = 360°$
To find a missing angle: $\text{missing angle} = 360° - \text{sum of known angles}$
Special quadrilateral rules: Square & Rectangle → all 90°; Parallelogram & Rhombus → opposite angles equal, adjacent angles supplementary; Kite → one pair of opposite angles equal.
What is the sum of interior angles of any quadrilateral?
A quadrilateral has angles 90°, 120°, and 75°. What is the fourth angle?
In a parallelogram, one angle is 65°. What is the size of the adjacent (co-interior) angle?
A quadrilateral has angles $4x°, 4x°, 2x°, 2x°$. What is the size of the largest angle?
Which statement about the angles of a kite is correct?
Q6. Quadrilateral $PQRS$ has $\angle P = 105°$ and $\angle R = 80°$. Given that $\angle Q = \angle S$, find the size of $\angle Q$ and $\angle S$.
Q7. Explain, using triangles, why the interior angle sum of a quadrilateral must be 360°. Your explanation should mention what a diagonal does and reference the angle sum of a triangle.
Q8. Design a quadrilateral whose four interior angles are in the ratio $1:2:3:4$. Calculate each angle and name what type of quadrilateral could have that angle combination (if any).
$\angle P + \angle Q + \angle R + \angle S = 360°$
$105 + \angle Q + 80 + \angle Q = 360$
$2\angle Q = 360 - 185 = 175$
$\angle Q = \angle S = 87.5°$
Drawing diagonal $AC$ divides quadrilateral $ABCD$ into triangles $\triangle ABC$ and $\triangle ACD$. Each triangle's angles sum to 180°. The angles of both triangles together make up all four interior angles of the quadrilateral. So the total interior angle sum = 180° + 180° = 360°.
Ratio $1:2:3:4$ → parts total = 10. Each part = $360° \div 10 = 36°$.
Angles: $36°, 72°, 108°, 144°$.
Check: $36+72+108+144 = 360°$ ✓
No standard named quadrilateral has exactly these angles, but it is a valid irregular quadrilateral (possibly a trapezium if one pair of sides is parallel).
A quadrilateral has four interior angles in arithmetic progression (AP). The smallest angle is 60°. Find all four angles and state what the common difference must be.