# Angles of a Quadrilateral - Formula and Examples

## What Are The Angles Of A Quadrilateral?

The **angles of a quadrilateral** are the four interior angles formed at its corners (vertices), and their sum is always **360°**. A **quadrilateral** is any closed shape with four straight sides and four vertices - a square, rectangle, [parallelogram](/content/math/geometry/angles-of-a-parallelogram/index.html), trapezium, or an irregular four-sided figure. Whatever the shape, the four interior angles add to 360°.

An **interior angle** is the angle measured inside the shape at a vertex, between two sides that meet there. The value of any single interior angle can range from just above 0° up to just below 360° (in a concave, or "dented", quadrilateral), but the four of them together are fixed at a full turn.

## Why Does The Sum Come Out To 360°?

Here is where the rule earns its keep, and it is worth deriving once rather than memorising. The angle sum of a **triangle** is 180° - that is the fact everything rests on. Now draw one diagonal across a quadrilateral, say from vertex A to vertex C. That single line cuts the four-sided figure into two triangles that share the diagonal.

- **Triangle ABC** carries 180° of angle.
- **Triangle ACD** carries another 180° of angle.
- Together they account for every interior angle of the original quadrilateral, with nothing left over and nothing counted twice.

So the total is 180° + 180° = **360°**. This is a special case of the general polygon rule, where an n-sided polygon has an interior-angle sum of (n−2)×180°. Put n=4 and you get (4−2)×180°=360° - the same answer, from the same triangle-splitting idea.

## Examples Of Angles Of A Quadrilateral

Six worked cases, moving from a clean missing-angle problem to cyclic and exterior-angle work. Each problem statement is bolded; the working is not.

### Example 1

**Three angles of a quadrilateral measure 85°, 90°, and 65°. Find the fourth angle.**  
Add the three known angles:  
85° + 90° + 65° = 240°  
Subtract from the total:  
360° − 240° = 120°  
Final answer: the fourth angle is **120°**.

### Example 2

**A student is told three angles of a quadrilateral are 90°, 90°, and 90°, and reasons that a four-cornered shape "must be a square, so the last angle is 90°." Check whether that reasoning is safe.**  
Do the sum instead:  
90° + 90° + 90° = 270°  
360° − 270° = 90°  
The answer does come out 90° here - but only because the numbers happened to. Change the third angle to 100° and the shape is no longer a rectangle at all; the fourth angle would be 360° − 280° = 80°. The correct method is to add the known angles and subtract from 360° every time, never to guess the shape and read the angle from its name.  
Final answer: **90°**, found by arithmetic, not by naming the shape.

### Example 3

**Two angles of a quadrilateral are equal, and the other two are 110° and 130°. Find the two equal angles.**  
Let each equal angle be x. The four angles sum to 360°:  
 x + x + 110° + 130° = 360°  
2x + 240° = 360°  
2x = 120°  
x = 60°  
Final answer: each equal angle is **60°**.

### Example 4

**The interior angle at one vertex of a quadrilateral is 108°. Find the exterior angle at that vertex.**  
An **exterior angle** is the angle between one side and the extension of the side next to it; it forms a straight line with the interior angle, so the two add to 180°.  
Exterior angle = 180° − 108° = 72°  
Final answer: the exterior angle is **72°**.

### Example 5

**A cyclic quadrilateral (all four vertices lie on a circle) has one angle of 95°. Find the angle opposite it.**  
In a **cyclic quadrilateral**, opposite angles are supplementary - they sum to 180°.  
Opposite angle = 180° − 95° = 85°  
Final answer: the opposite angle is **85°**.

### Example 6

**The four angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find each angle.**  
Let the angles be x, 2x, 3x, and 4x. Their sum is 360°:  
 x + 2x + 3x + 4x = 360°  
10x = 360°  
 x = 36°  
So the angles are:  
36°, 72°, 108°, 144°  
Final answer: **36°, 72°, 108°, and 144°** - and as a check, they add to 360°.

## Where The 360° Rule Shows Up

The angle-sum property is not a classroom curiosity. Surveyors closing a four-sided plot of land check that the measured corner angles total 360°; a shortfall signals a measurement error before the boundary is filed. Kite-makers and sail-cutters rely on it to keep a four-sided panel flat rather than warped. In coordinate geometry, the rule lets you verify that four plotted points genuinely form a [quadrilateral](/content/math/geometry/quadrilaterals/index.html). The theme is always the same: four corners, one full turn, and a way to catch a mistake before it becomes expensive.

Historically, this is one of the oldest results in plane geometry, following directly from Euclid's triangle angle-sum theorem in the _Elements_ (around 300 BCE). You can read a short account of the [triangle angle sum](https://en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle) that everything here is built on.

## Common Mistakes With Angles Of A Quadrilateral

### Mistake 1: Stopping at the sum instead of subtracting

**Where it slips in:** Missing-fourth-angle problems, when three angles are given.

**Don't do this:** Add the three known angles and report that total as the answer. The first instinct on these problems is to add the given angles and stop there - the subtraction from 360° is the step that gets dropped.

**The correct way:** Add the known angles, then subtract that sum from 360°. If three angles give 240°, the missing angle is 360° − 240° = 120°, not 240°.

### Mistake 2: Using 180° instead of 360°

**Where it slips in:** Right after studying triangles, when the 180° figure is still fresh.

**Don't do this:** Assume the four angles sum to 180° because triangles do. The habit of reaching for 180° carries over from triangle work and lands on the wrong figure.

**The correct way:** Remember the shape splits into _two_ triangles, so the sum is 2×180° = 360°. Deriving it once from the diagonal split means you never confuse the two totals under exam pressure.

### Mistake 3: Confusing interior and exterior angles

**Where it slips in:** Problems that switch between the angle inside the shape and the angle formed by extending a side.

**Don't do this:** Read a given exterior angle as if it were the interior angle, or subtract from 360° when the problem asked for an exterior angle.

**The correct way:** An interior and its exterior angle together make a straight line - they sum to 180°, not 360°. Label which one the problem gives before you compute.

## Conclusion

- The four **interior angles of a quadrilateral** always sum to **360°**.
- The rule comes from splitting the shape into two triangles: 2×180°=360°.
- To find a missing angle, add the known angles and subtract from 360°.
- An interior and its exterior angle sum to 180°, not 360°.
- In a cyclic quadrilateral, opposite angles are supplementary (sum to 180°).