Before talking about the quadrilateralsangle sum property, let us recall what angles and quadrilateral is. The angle is formed when two line segment joins at a single point. An angle is measured in degrees (°). Quadrilateral angles are the angles formed inside the shape of a quadrilateral. The quadrilateral is four-sided polygon which can have or not have equal sides. It is a closed figure in two-dimension and has non-curved sides. A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles andthe sum of all the angles is 360°. When we draw a draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180°. Therefore, the total angle sum of the quadrilateral is 360°. Angle sum is one of the properties of quadrilaterals. In this article, w will learn the rules of angle sum property.
| Quadrilateral | Area Of Quadrilateral |
| Construction Of Quadrilaterals | Types Of Quadrilaterals |
Angle Sum Property of a Quadrilateral
According to the angle sum property of a Quadrilateral, the sum of all the four interior angles is 360 degrees.
Proof: In the quadrilateral ABCD,
- ∠ABC, ∠BCD, ∠CDA, and ∠DAB are the internal angles.
- AC is a diagonal
- AC divides the quadrilateral into two triangles, ∆ABC and ∆ADC
We have learned that the sum of internal angles of a quadrilateral is 360°, that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.
let’s prove that the sum of all the four angles of a quadrilateral is 360 degrees.
- We know that the sum of angles in a triangle is 180°.
- Now consider triangle ADC,
∠D + ∠DAC + ∠DCA = 180° (Sum of angles in a triangle)
- Now consider triangle ABC,
∠B + ∠BAC + ∠BCA = 180° (Sum of angles in a triangle)
- On adding both the equations obtained above we have,
(∠D + ∠DAC + ∠DCA) + (∠B + ∠BAC + ∠BCA) = 180° + 180°
∠D + (∠DAC + ∠BAC) + (∠BCA + ∠DCA) + ∠B = 360°
- We see that (∠DAC + ∠BAC) = ∠DAB and (∠BCA + ∠DCA) = ∠BCD.
- Replacing them we have,
∠D + ∠DAB + ∠BCD + ∠B = 360°
∠D + ∠A + ∠C + ∠B = 360°.
Or, the sum of angles of a quadrilateral is 360°. This is the angle sum property of quadrilaterals.
Quadrilateral Angles
A quadrilateral has 4 angles. The sum of its interior angles is 360 degrees. We can find the angles of a quadrilateral if we know 3 angles or 2 angles or 1 angle and 4 lengths of the quadrilateral. In the image given below, a Trapezoid (also a type of Quadrilateral) is shown.
The sum of all the angles∠A +∠B +∠C +∠D = 360°
In the case of square and rectangle, the value of all the angles is 90 degrees. Hence,
∠A = ∠B = ∠C = ∠D = 90°
A quadrilateral, in general, has sides of different lengths and angles of different measures. However, squares, rectangles, etc. are special types of quadrilaterals with some of their sides and angles being equal.
Do the Opposite side in a Quadrilateral equals 180 Degrees?
There is no relationship between the opposite side and the angle measures of a quadrilateral. To prove this, the scalene trapezium has the side length of different measure, which does not have opposite angles of 180 degrees. But in case of some cyclic quadrilateral, such as square, isosceles trapezium, rectangle, the opposite angles are supplementary angles. It means that the angles add up to 180 degrees. One pair of opposite quadrilateral angles are equal in the kite and two pair of the opposite angles are equal in the quadrilateral such as rhombus and parallelogram. It means that the sum of the quadrilateral angles is equal to 360 degrees, but it is not necessary that the opposite angles in the quadrilateral should be of 180 degrees.
Types of Quadrilaterals
There are basically five types of quadrilaterals. They are;
- Parallelogram: Which has opposite sides as equal and parallel to each other.
- Rectangle: Which has equal opposite sides but all the angles are at 90 degrees.
- Square: Which all its four sides as equal and angles at 90 degrees.
- Rhombus: Its a parallelogram with all its sides as equal and its diagonals bisects each other at 90 degrees.
- Trapezium: Which has only one pair of sides as parallel and the sides may not be equal to each other.
Example
1. Find the fourth angle of a quadrilateral whose angles are 90°, 45° and 60°.
Solution: By the angle sum property we know;
Sum of all the interior angles of a quadrilateral = 360°
Let the unknown angle be x
So,
90° + 45° + 60° + x = 360°
195° + x = 360°
x = 360° – 195°
x = 165°
To learn more about quadrilaterals and their properties, download BYJU’S-The Learning App.
FAQs
A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles and the sum of all the angles is 360°. When we draw a draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180°. Therefore, the total angle sum of the quadrilateral is 360°.
How do you prove the angle sum property? ›
Proof of the Angle Sum Property
Step 1: Draw a line PQ that passes through the vertex A and is parallel to side BC of the triangle ABC. Step 2: We know that the sum of the angles on a straight line is equal to 180°. In other words, ∠PAB + ∠BAC + ∠QAC = 180°, which gives, Equation 1: ∠PAB + ∠BAC + ∠QAC = 180°
How do you prove the angle sum property of a quadrilateral is 360? ›
Proof: In the quadrilateral ABCD, ∠ABC, ∠BCD, ∠CDA, and ∠DAB are the internal angles. We have learned that the sum of internal angles of a quadrilateral is 360°, that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.
What is the sum of the angles in a quadrilateral theorem? ›
Quadrilaterals are composed of two triangles. Seeing as we know the sum of the interior angles of a triangle is 180°, it follows that the sum of the interior angles of a quadrilateral is 360°.
What is the state and prove the angle sum property of a quadrilateral? ›
A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles and the sum of all the angles is 360°. When we draw a draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180°. Therefore, the total angle sum of the quadrilateral is 360°.
How do you prove the angle theorem? ›
To prove this theorem, let's assume a pair of intersecting straight lines that form an angle A between them. Now, we know that any two points on a straight line form an angle of 180 degrees between them. So, for the given pair of lines, the remaining angles on both the straight lines would be 180 - A.
How do you prove that ABCD is a quadrilateral? ›
We can say that a quadrilateral is a closed figure with four sides : e.g. ABCD is a quadrilateral which has four sides AB, BC, CD and DA, four angles ∠A,∠B,∠C and ∠D and four vertices A, B, C and D and also has two diagonals AC and BD. i.e. A quadrilateral has four sides, four angles, four vertices and two diagonals.
What is the theorem 2 quadrilateral? ›
Theorem 2. The ratio between the diagonals and the sides can be defined and is known as Cyclic quadrilateral theorem. If there's a quadrilateral which is inscribed in a circle, then the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.
How many theorems are in quadrilateral? ›
1st Theorem- The diagonal divides the parallelogram into two congruent triangles. 2nd Theorem- The opposite side of a parallelogram are equal. 3rd Theorem- The quadrilaterals in which each pair of opposite sides are equal are called parallelogram. 4th Theorem- The opposite angles of a parallelogram are equal.
What are the rules for quadrilateral angles? ›
The quadrilateral is a parallelogram. Opposite angles are equal. Angles in a quadrilateral add up to 360° and opposite angles are equal.
For example, if we take a quadrilateral and apply the formula using n = 4, we get: S = (n − 2) × 180°, S = (4 − 2) × 180° = 2 × 180° = 360°. Therefore, according to the angle sum property of a quadrilateral, the sum of its interior angles is always 360°. Similarly, the same formula can be applied to other polygons.
How do you prove that the sum of all angles around a point is 360? ›
Let MP & NQ be lines intersecting at O. Adding the equations, Angle MON + Angle NOP + Angle MOQ + Angle QOP = 180 + 180 degrees = 360 degrees. Therefore all angles around O sum upto 360 degrees.
What is a quadrilateral and its properties? ›
A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices and four angles. It is formed by joining four non-collinear points. The sum of interior angles of quadrilaterals is always equal to 360 degrees.
What is the quadrilateral congruence theorem? ›
If they have a side together with the adjacent angles respectively congruent, then the quadrilaterals are congruent. Proof. In order to show that Q = (A, B, C, D) and Q/ = (A/,B/,C/,D/) are congruent, we may suppose that they have AB = A/B/, ˆA = ˆA/ and ˆB = ˆB/.
How do you prove angle addition postulates? ›
The Angle Addition Postulate states that the sum of two adjacent angle measures will equal the angle measure of the larger angle that they form together. The formula for the postulate is that if D is in the interior of ∠ ABC then ∠ ABD + ∠ DBC = ∠ ABC. Adjacent angles are two angles that share a common ray.
How do you prove angle congruence? ›
The complement theorem states that if angle A and angle B are both complementary to the same angle, then A and B are congruent. The supplement theorem states that if angle A and angle B are both supplementary to the same angle, then A and B are congruent.
How to prove that the sum of a triangle is 180? ›
Mark the angles ∠ 1 , ∠ 2 , ∠ 3 , ∠ 4 and as shown in the figure.
- STEP 2 : Proving that sum of the angles of a triangle is.
- ∠ 2 = ∠ 4 (Alternate interior angles) ...
- ∠ 3 = ∠ 5 (Alternate interior angles) ...
- Adding equation and.
- We know that angles on a straight line add up to.
- ∴ ∠ 1 + ∠ 4 + ∠ 5 = 180 °
- ⇒ ∠ 1 + ∠ 2 + ∠ 3 = 180 °
