Angles of Square
Angles of a square are all right angles. A square is a quadrilateral having the opposite sides parallel. The square shape is characterized by only one dimension, which is its side length. All four sides are equal and each of the angle measure 90°. Let us discuss more on the angles of a square and solve a few examples to understand the concept better.
|1.||What are Angles of Square?|
|2.||Sum of Angles of Square|
|3.||Opposite Angles of Square|
|4.||Diagonal Angles of Square|
|5.||FAQs on Angles of Square|
What are Angles of Square?
A square is a two-dimensional, closed figure with four sides and four corners. The length of all four sides is equal and the measure of all four angles is also equal. A square is characterized as a quadrilateral with all its angles measuring 90°. The properties of angles in a square are as follows:
- There are four interior angles, each angle is a right angle.
- The opposite angles of a square are of equal measure.
- Any two consecutive angles of a square are supplementary.
- The diagonal angles of a square bisect each other and form right angles at the center.
Sum of Angles of Square
A square is a quadrilateral only. As per the angle sum property of quadrilateral, the sum of all its interior angles is 360°. Also, we know that the four angles in a square, one at each vertex, are congruent. Thus, the interior angle of a square at each vertex is 360°/4 = 90°.
Opposite Angles of Square
A square has all four sides equal with the opposite sides being parallel to each other. For a pair of opposite sides of a square, their adjacent sides are transversals, which results in the property that the two consecutive angles are supplementary, that is, they add up to 180°.
The opposite angles in any quadrilateral are non-adjacent angles formed by two intersecting lines and we know that the adjacent sides of a square are perpendicular. Thus, the opposite angles of a square are equal and measure 90°.
Diagonal Angles of Square
The diagonals of a square are perpendicular to each other bisecting each other at 90°. Also, all four sides of a square are equal, say each side measures 's'. A diagonal divides a square into two congruent triangles, that too right triangles with their hypotenuse being the same. Also, each diagonal divides its corresponding vertex angles into two congruent angles, that is angles of the same measure (45° each).
Each diagonal forms the hypotenuse of the right triangles so formed. Thus, applying the Pythagoras theorem, taking square root on both sides gives, √(d2) = √( 2a2), the length of the diagonal with sides s is √2 × s. Thus, the diagonal of a square formula is Diagonal of Square (d) = √2 × a. Thus, the length of the diagonals is equal.
- A square can also be referred to as a rectangle with two opposite sides having an equal length.
- The interior angles of a square are all equal and sum up to 360°.
- The diagonals of a square make right angles at the center.
Angles of Square Examples
Example 1: ABCD is a square. Find the angle x.
Given: Square PQRS
We know that the angles of a square are all right angles and the diagonal bisects the vertex angle into two congruent angles.
∠QRS = 90°. Thus, ∠QRP = ∠SRP = ∠x = 90/2 = 45° (diagonal PR bisects ∠R)
Therefore, ∠x = 45° .
Example 2: Using properties of angles of the square, find the diagonal of a square whose side length is 5 units.
Given: The side of a square is = 5 units
According to the properties of angles of the square, the diagonal of a square = (d) = √2 × a
Length of diagonal of square = √2 × 5 = 7.05 units.
Therefore, the length of the diagonal of a square whose side length is 5 units is 7.05 units.
Example 3: Find the length of the diagonal of a square if its area is 36 square units.
Area of the square = 36 square units
We know that the area of a square = a2 = 36
Therefore, a = 6 units
We know that the diagonal of a square formula is, d = a√2
Therefore, the length of the diagonal = a√2 = 6√2 = 8.49 units
Therefore, the length of the diagonal of the square = 8.49 units.
FAQs on Angles of Square
What are the Angles of Square in Geometry?
A square has four equal sides and four equal angles. It has two diagonals, each of which splits a corner into two equal angles. The diagonals of a square also act as its lines of symmetry. Thus, a square has four interior angles, each of which is equal to 90°, whereas its diagonals also make right angles at the center as they are perpendicular to each other.
What is the Sum of the Interior Angles of Square?
A square is a special type of quadrilateral having all its sides equal. Thus, being a quadrilateral, the sum of its interior angles equals 360°.
What is the Measure of Each of the Angles of a Square?
Since all the sides of a square and its angles are equal, with all its adjacent sides being perpendicular, thus, each of its angles measures 90°. The diagonals also are equal in length and bisect each other, making right angles at the center.
How Do you Find an Angle in a Square?
A square is a regular polygon having a flat shape, with its four sides, all equal in length and four angles, all equal in measure. For any regular polygon, the formula to find the sum of the measure of the interior angles is (n - 2) * 180. Thus, knowing that all the angles are equal, the measure of an interior angle = [(n - 2) * 180] / n = [(4-2)* 180] / 4 = 360/4 = 90°.
Also, since it's a quadrilateral and thus has the sum of interior angles equal to 360°. Thus, the measure of one interior angle = 360/4 = 90°, and because a square has all its angles equal in measure, thus each angle equals 90°.
What is the Sum of the Exterior Angles of a Square?
As per the exterior angle property of polygons, the sum of exterior angles in a polygon equals 360 degrees. Thus, in the case of any equiangular polygon, the measure of an exterior angle = 360/n, where n is the number of sides in the polygon. Therefore, the sum of exterior angles of a square equals 360°.