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°
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FAQs
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.
What is the proof of 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°.
What is the quadrilateral theorem and proof? ›
Theorem 1: A quadrilateral is a parallelogram if both pairs of opposite sides are congruent. Theorem 2: A quadrilateral is a parallelogram if both pairs of opposite angles are congruent. Theorem 3: A quadrilateral is a parallelogram if its diagonals bisect each other.
What is the sum of the angles of 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°.
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°.
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°
What are the statements for quadrilateral proof? ›
Proving a Quadrilateral is a Parallelogram
To prove that a quadrilateral is a parallelogram, show that it has any one of the following properties: • Both pairs of opposite sides are parallel. Both pairs of opposite angles are congruent. Both pairs of opposite sides are congruent. Diagonals bisect each other.
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 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/.
What is theorem 8.5 if in a quadrilateral? ›
Theorem 8.5 : If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
Properties of the quadrilaterals – An overview
| Properties of quadrilaterals | Rectangle | Parallelogram |
|---|
| All angles are equal | Yes | No |
| Opposite angles are equal | Yes | Yes |
| Sum of two adjacent angles is 180 | Yes | Yes |
| Bisect each other | Yes | Yes |
4 more rowsConjecture (Quadrilateral Sum ): The sum of the measures of the interior angles in any convex quadrilateral is 360 degrees. Proof: The sum of the measures of the interior angles of any quadrilateral can be found by breaking the quadrilateral into two triangles.
What is the proof of angle sum property of quadrilateral? ›
∠BCA + ∠DCA = ∠BCD. That is, ∠D + ∠C + ∠A + ∠B = 360°. ∠A + ∠B + ∠C + ∠D = 360°. Hence the angle sum property of the quadrilateral theorem proved.
What is the angle theorem for a quadrilateral? ›
The interior angles of a quadrilateral add up to 360°. This value is calculated from the formula given by the angle sum property of polygons. Sum of interior angles = (n − 2) × 180°, where 'n' represents the number of sides of the given polygon. In this case, n = 4.
What is the formula for the sum of the exterior angles of a quadrilateral? ›
Each vertices consists of an internal angle and an external angle. The sum of the external angles is equal to 360-x + 360-y + 360- w +360-z. But x+y+w+z is equal to 360 degrees, so the sum of the external angles 4*360- (x+y+w+z) or 4*360–360, which is equal to 3*360 or 1080.
What is the proof properties of angles? ›
The properties of angles are applied in situations that are often referred to as angle proofs. These are theorems that include angle analysis, such as in the statement 'if a straight segment A is intersected by a straight segment B, the sum of the two angles formed between A and B is 180 o . '
What is the verify angle sum property for a concave quadrilateral? ›
We know that, the sum of interior angles of any polygon (convex or concave) having n sides is(n -2) × 180°. Therefore, the sum of angles of a concave quadrilateral is equal to 360°.
How do you prove angles are congruent in a quadrilateral? ›
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/.
What are the angle properties of a quadrilateral? ›
Every quadrilateral has 4 vertices, 4 angles, and 4 sides. The total of its interior angles = 360 degrees.
