# Angles of a Rectangle - Properties and Examples ## TL;DR Every one of a rectangle's four interior angles is 90°, and they add to 360°. This article explains why all four are right angles, what the diagonals do to those corners (they split each 90° into two acute angles and are equal but not perpendicular), and works through six examples plus the mistakes students make. ## The Shape So Reliable We Build Cities On It Look up from this screen and count the right angles around you: the door, the window, the wall, the book, the tabletop. The rectangle is the most common shape in the built world for one reason - its four equal, square corners make it stack, tile, and align perfectly. Those four **90° angles** are not a coincidence of the drawing; they are the definition of the shape, and understanding _why_ they must all be right angles is the whole of this topic. ## What Are The Angles Of A Rectangle? A **rectangle** is a four-sided flat shape (a [quadrilateral](/content/math/geometry/quadrilaterals/index.html)) in which **all four interior angles are right angles**, each measuring exactly **90°**. Because there are four of them, the interior angles of a rectangle always **sum to 360°**. That is the answer in full. Every corner is 90°; the four corners total 360°. A quick term first: an **interior angle** is the angle formed _inside_ the shape between two sides that meet at a corner (a **vertex**). Using angle notation, if the rectangle is ABCD then ∠A=∠B=∠C=∠D=90°. **Why are all four angles 90° and not just some?** Because a rectangle is defined that way, and the definition is self-consistent. Any four-sided shape has interior angles summing to 360°. A rectangle is a [parallelogram](/content/math/geometry/properties-of-a-rectangle/index.html) with one right angle - and in a parallelogram, opposite angles are equal and adjacent angles add to 180°. Fix one angle at 90° and the rules force all four to 90°: - The angle opposite it is equal, so it is 90° too. - The two adjacent angles must make 180° with a 90° angle, so each is 180°−90°=90°. All four land on 90° with no freedom left. That derivation, not the picture, is why the corners are square. ## What The Diagonals Do To The Angles Draw the two **diagonals** - the lines joining opposite corners - and a second layer of angles appears. This is where a rectangle behaves differently from a square, so it is worth being precise. - **A diagonal splits each 90° corner into two acute angles that add to 90°.** The diagonal is not an angle bisector unless the rectangle happens to be a square, so those two pieces are usually _unequal_. - **The two diagonals are equal in length.** This is a genuine property of every rectangle. - **The diagonals bisect each other** (they cut each other in half at the centre) **but do not meet at right angles.** At the centre they form two pairs of angles that are generally _not_ 90°. That last point is the key contrast with the square. In a square, the diagonals _are_ perpendicular; in a rectangle they are not, unless the rectangle is a square. Equal diagonals: yes. Perpendicular diagonals: no. ## Examples Of Angles Of A Rectangle ### Example 1 **Find the sum of all four interior angles of a rectangle.** Each interior angle is 90°, and there are four. 90° + 90° + 90° + 90° = 360° Final answer: 360°. ### Example 2 **In rectangle ABCD, a diagonal makes an angle of 35° with one side at corner A. A student says the other part of that corner is also 35°. Is that right?** The instinct is that a diagonal "splits the corner evenly," so both pieces look like they should match at 35°. Watch where it breaks: the two pieces of a right-angle corner must add to 90°, the full corner. If both were 35°, they would total 70°, not 90°. So they cannot both be 35° unless the diagonal is a bisector, which it is not in a general rectangle. The correct way uses ∠1 + ∠2 = 90°: ∠2 = 90° − 35° = 55° Final answer: the other part of the corner is 55°, not 35°. ### Example 3 **One interior angle of a rectangle is given as (2x + 10)°. Find x.** Every interior angle of a rectangle is 90°. 2x + 10 = 90 2x = 80 x = 40 Final answer: x = 40. ### Example 4 **In rectangle ABCD, diagonal AC makes a 28° angle with side AB. Find the angle it makes with side AD at the same corner.** At corner A, sides AB and AD meet at 90°, and the diagonal splits that corner. ∠(diagonal, AD) = 90° − 28° = 62° Final answer: 62°. ### Example 5 **The diagonals of a rectangle meet at the centre. One of the four angles at the centre is 110°. Find the other three.** The diagonals cross, so the four centre angles are two pairs of vertically opposite (equal) angles, and each adjacent pair is supplementary (adds to 180°). 180° − 110° = 70° So the four angles at the centre are 110°, 70°, 110°, 70°. Note none is 90°, confirming the diagonals are not perpendicular here. Final answer: 110°, 70°, 110°, 70°. ### Example 6 **A diagonal of a rectangle divides it into two triangles. In one triangle, the diagonal makes a 40° angle with the base. Find the third angle of that triangle.** Each triangle formed by a diagonal has one 90° angle (a corner of the rectangle). The three angles of any triangle sum to 180°. 40° + 90° + θ = 180° θ = 50° Final answer: the third angle is 50°. ## Why The Right Angle Rules The Built World: "Squareness Is Buildability" The reason rectangles dominate architecture is not aesthetic - it is structural. Right angles let materials be cut, stacked, and joined without gaps, and they let builders _check their work_ with a simple tool. - **Bricks and tiles** are rectangles because 90° corners tessellate perfectly, leaving no wasted space and no weak seams. - **The 3-4-5 check.** Builders confirm a corner is truly square by measuring 3 units along one wall, 4 along the other, and checking the diagonal is 5 - a right angle if and only if the numbers fit, which is the [converse of the Pythagoras theorem](/content/math/geometry/converse-of-pythagoras-theorem/index.html) at work on a job site. - **Equal diagonals as a level check.** Because a rectangle's diagonals are equal, carpenters measure both diagonals of a frame; if they match, the frame is a true rectangle and not a leaning parallelogram. ## Tripping Points To Avoid ### Mistake 1: Assuming the diagonal bisects the corner angle **Where it slips in:** any problem where a diagonal cuts a rectangle's corner. **Don't do this:** split the 90° corner into two equal 45° halves. **The correct way:** the diagonal only bisects the corner in a _square_. In a general rectangle the two pieces are unequal but still add to 90°. The first instinct is to see "diagonal through a corner" and assume symmetry; unless the sides are equal, that symmetry is not there. Use ∠1 + ∠2 = 90°. ### Mistake 2: Thinking the diagonals meet at 90° **Where it slips in:** confusing a rectangle with a square or rhombus. **Don't do this:** mark a right angle where the diagonals cross. **The correct way:** a rectangle's diagonals are _equal_ and _bisect each other_, but they are **not** perpendicular. Perpendicular diagonals belong to the square and the rhombus. The confusion between "equal diagonals" and "perpendicular diagonals" is the single most common rectangle error - they are different properties, and a rectangle has only the first. ### Mistake 3: Forgetting the interior angles sum to 360°, not 180° **Where it slips in:** carrying over the triangle rule to a four-sided shape. **Don't do this:** claim the four angles add to 180° because triangles do. **The correct way:** a triangle's angles sum to 180°; a quadrilateral's sum to 360°, one full turn. A rectangle's four 90° angles giving 360° is a clean check on this. ## Conclusion - All four **interior angles of a rectangle are 90°**, summing to 360°. - The four right angles follow from the definition: a rectangle is a parallelogram with one right angle, which forces all four. - The diagonals are **equal** and **bisect each other** but are **not perpendicular** - that is the key difference from a square. - A diagonal splits each 90° corner into two acute angles that add to 90°, and only bisects the corner when the rectangle is a square.