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, 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 $(n-2) \times 180°$. Put $n = 4$ and you get $(4-2) \times 180° = 360°$ - the same answer, from the same triangle-splitting idea.
$$\text{Sum of interior angles} = (n-2)\times 180° = (4-2)\times 180° = 360°$$
Where each symbol lands: $n$ is the number of sides, and $(n-2)$ counts how many triangles the diagonals cut the shape into. The "why" is not decoration - it is what lets you handle a five-sided or ten-sided figure the day you meet one.
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.
The tempting move is to name the shape first, then read the angle off the name. Watch it break: the name "square" was never given, only three right angles were. If you assume square, you skip the arithmetic and lock in 90° by habit.
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°.
$$\text{Exterior angle} = 180° - 108° = 72°$$
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°. This is a property of quadrilaterals inscribed in a circle, tied to the inscribed-angle theorem.
$$\text{Opposite angle} = 180° - 95° = 85°$$
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:
$$36°, \quad 72°, \quad 108°, \quad 144°$$
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. 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 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 $2 \times 180° = 360°$. 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: $2 \times 180° = 360°$.
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°).
To take angles of a quadrilateral further with a teacher, explore Bhanzu's geometry tutor or middle school math tutor sessions, or browse math classes online.
A Practical Next Step
Practice these problems to solidify your understanding. Try finding the fourth angle when three are 100°, 75°, and 95° (Answer to Question 1: 90°), then find each angle of a quadrilateral whose angles are in the ratio 2 : 3 : 4 : 6 (Answer to Question 2: 48°, 72°, 96°, 144°). If you get stuck on the subtraction step, return to the Examples section above. Want a live Bhanzu trainer to walk through more problems? Book a free demo class -
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