Any four-sided shape (quadrilateral) can be split into two triangles by drawing a diagonal. Each triangle has angles summing to 180°, so the quadrilateral's four angles sum to \(2 \times 180° = 360°\).
Example 1 — Find the missing angle x
A quadrilateral has angles 110°, 85°, 70°, and \(x\). Find \(x\).
Step 1
All angles in a quadrilateral sum to 360°: \(110° + 85° + 70° + x = 360°\)
Step 2
Add the known angles: \(110 + 85 + 70 = 265°\)
Step 3
Subtract: \(x = 360° - 265° = \mathbf{95°}\)
Example 2 — Four equal angles
A quadrilateral has four equal angles. Find each angle.
Step 1
Let each angle be \(x\). Then \(4x = 360°\)
Step 2
\(x = 360° \div 4 = \mathbf{90°}\)
Step 3
Each angle is 90° — this shape is a rectangle or a square.
Remember: Any quadrilateral can be split into 2 triangles by drawing a diagonal. Since each triangle has angles summing to 180°, the total is \(2 \times 180° = 360°\).
✓ Quick Check 1 — Angles Summing to 360°
Question 1 of 10
Properties of Special Quadrilaterals
Different types of quadrilateral have special angle properties that let us find missing angles even faster than using the 360° rule alone.
Example 1 — Parallelogram with one known angle
A parallelogram has one angle of 65°. Find the other three angles.
Step 1
Opposite angles in a parallelogram are equal. The angle opposite 65° is also 65°.
Step 2
Adjacent angles are co-interior (sum to 180°). \(180° - 65° = \mathbf{115°}\)
Step 3
Four angles: 65°, 115°, 65°, 115°. Check: \(65+115+65+115=360°\) ✓
Example 2 — Trapezium with co-interior angles
A trapezium has co-interior angles 80° and \(x\) on the same parallel side. Find \(x\).
Step 1
Co-interior angles (between parallel lines, same side) sum to 180°.
Step 2
\(80° + x = 180°\)
Step 3
\(x = 180° - 80° = \mathbf{100°}\)
Rectangle: all angles 90°. Parallelogram: opposite angles equal; adjacent angles supplementary (sum to 180°).
Watch out: A rhombus looks like a squashed square. Its angles are NOT all 90° unless it is actually a square. A rhombus has equal opposite angles and adjacent angles summing to 180° — same rules as a parallelogram.
✓ Quick Check 2 — Properties of Special Quadrilaterals
Question 1 of 10
Finding Missing Angles Using Quadrilateral Properties
We combine the 360° rule with specific shape properties to find missing angles efficiently. Always identify the type of quadrilateral first.
Example 1 — Missing angle in a kite
A kite has angles 110°, 70°, 70°, and \(x\). Find \(x\).
Step 1
All angles in a quadrilateral sum to 360°: \(110° + 70° + 70° + x = 360°\)
Step 2
Add known angles: \(110 + 70 + 70 = 250°\)
Step 3
\(x = 360° - 250° = \mathbf{110°}\)
Step 4
This makes sense — a kite is symmetric, so the two angles between unequal sides (70°, 70°) are equal, and both tip angles here are 110°.
Example 2 — Reflex angle at an internal corner
An L-shaped room has an internal corner. The exterior angle at that corner is 90°. Find the interior (reflex) angle.
270° is a reflex angle (greater than 180°) — the concave corner of the L-shape.
For kites: the two angles between the unequal sides are always equal. The axis of symmetry splits the kite into two congruent triangles — this forces those angles to match.
Use quadrilateral properties to find each missing angle. Type your answer as a number (degrees), then click Check All.
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Question
Answer
Result
1
Angles: 90°, 90°, 90°, ?
2
Parallelogram, one angle = 75° — find the adjacent angle
3
Angles: 80°, 100°, 95°, ?
4
Rhombus, one angle = 60° — find the opposite angle
5
Angles: 110°, 110°, 70°, ?
6
Rectangle: each angle = ?
7
Angles: 100°, 80°, 75°, ?
8
Parallelogram, one angle = 120° — find the adjacent (smaller) angle
9
Angles: 75°, 105°, 105°, ?
10
Rhombus, one angle = 130° — find the adjacent angle
⚠ Things to Watch Out For
Using 180° instead of 360°: Interior angles of a quadrilateral sum to 360° (not 180°). 180° applies to triangles.
Assuming all quadrilaterals have a right angle: Only rectangles, squares, and right-angled trapeziums have 90° angles. Parallelograms, rhombuses, and kites generally do not.
Parallelogram angle properties: In a parallelogram, opposite angles are EQUAL and adjacent angles are supplementary (add to 180°). Don't assume all angles are equal.
Trapezium — co-interior angles: In a trapezium with two parallel sides, the co-interior angles (between the parallel sides on the same side) add to 180°.
Irregular quadrilaterals: For irregular quadrilaterals, use the 360° rule and set up an equation to find unknown angles. Don't guess based on appearance.