The angles of a quadrilateral are in ratio 1:2:3:4 — find each angle

Question

The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find all four angles.

Solution — Step by Step

The sum of all interior angles of any quadrilateral is always 360°. This comes from the fact that any quadrilateral can be split into two triangles, each with an angle sum of 180°.

So: $\angle A + \angle B + \angle C + \angle D = 360°$

The angles are in ratio 1 : 2 : 3 : 4. Let the common factor be $k$ . Then:

\angle A = k, \quad \angle B = 2k, \quad \angle C = 3k, \quad \angle D = 4k

k + 2k + 3k + 4k = 360°

10k = 360°

k = 36°

\angle A = 1 \times 36° = 36°

\angle B = 2 \times 36° = 72°

\angle C = 3 \times 36° = 108°

\angle D = 4 \times 36° = 144°

Verification: $36° + 72° + 108° + 144° = 360°$ ✓

Why This Works

When angles are given in a ratio, the ratio tells us the relative sizes — not the actual sizes. The variable $k$ acts as the “scale factor.” Since we have an additional constraint (the sum must equal 360°), we can find $k$ uniquely.

This technique applies to any polygon: set angles as multiples of $k$ , use the angle sum property to find $k$ , then scale up.

Alternative Method

You can also think of it as: the four angles take up fractions $\frac{1}{10}$ , $\frac{2}{10}$ , $\frac{3}{10}$ , $\frac{4}{10}$ of the total 360°.

\angle A = \frac{1}{10} \times 360° = 36°, \quad \angle B = \frac{2}{10} \times 360° = 72°

\angle C = \frac{3}{10} \times 360° = 108°, \quad \angle D = \frac{4}{10} \times 360° = 144°

To quickly find individual angles from a ratio when the total is known: each angle = (its ratio part ÷ sum of all ratio parts) × total. Here: each angle = (ratio part ÷ 10) × 360°.

Common Mistake

Students sometimes use the wrong angle sum property. This question is about a quadrilateral (4 sides) — angle sum = 360°. If the problem were about a triangle, the sum would be 180°. Always identify the polygon first. For a polygon with $n$ sides, the interior angle sum is $(n-2) \times 180°$ .

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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