In this chapter we will about about the angles of quadrilateral, also the method to find the missing angle if measure of other angles are given.
What are angles in a quadrilateral ?
Here we are talking about the interior angles of quadrilateral.
It’s the interior angle formed between the two adjacent sides. In Quadrilateral, there are 4 interior angles present.
Consider the above quadrilateral ABCD in which ∠1, ∠2, ∠3 and ∠4 are the interior angles of quadrilateral.
∠1 is formed between side DA & AB
∠2 is formed between side CB & AB
∠3 is formed between side BC & DC
∠4 is formed between side AD & CD
Sum of angles of Quadrilateral
The sum of all four angles of quadrilateral measures 360 degree.
Please remember this as a rule as it would help you solve lots of geometry related problems.
Let’s again consider the same quadrilateral ABCD.
Using the angle sum property rule of quadrilateral we can write;
∠1+ ∠2 + ∠3 + ∠4 = 360 degree
Proof of angle sum property of quadrilateral
Take the above quadrilateral ABCD and join one of its diagonal AC.
Now when we join the diagonal AC, we get two triangles ABC and ADC.
We know that sum of all angles of triangle measures 180 degree.
Sum ( all angles of triangle ABC ) = 180 degree
Sum ( all angles of triangle CDA ) = 180 degree
Now if we add angles of both the triangle we will get to total of 360 degrees.
Sum ( triangle ABC + triangle CDA ) = 180 + 180
Sum ( quadrilateral ABCD) = 360
Angle property of different quadrilateral
Angle Property of Square
In square, all interior angles are right angles.
Angle property of rectangle
In rectangle, all interior angles are right angle (i.e. 90 degree)
Angle property of Parallelogram
In parallelogram, opposite angles are equal to each other.
∠A = ∠C
∠D = ∠B
Also, sum of supplementary angles measure 180 degree
∠A + ∠B = 180 degree
∠B + ∠C = 180 degree
∠C + ∠D = 180 degree
∠D + ∠A = 180 degree
Angle property of Rhombus
The angle property of rhombus is same as that of parallelogram.
i.e. opposite angles are equal and sum of consecutive angles measure 180 degree.
One additional property of rhombus is that its diagonal bisect each other at 90 degrees.
i.e. ∠AOB = 180 degree
Angle property of Trapezium
In trapezium, one pair of sides are parallel to each other.
We know that when parallel sides are intersected by transversal then sum of interior angles measures 180 degree.
Hence,
∠A + ∠D = 180 degree
∠B + ∠C = 180 degree
How to find missing angle in quadrilateral ?
If all the angle of quadrilateral is known except one, you can find the missing angle using angle sum property of quadrilateral.
Just subtract all the know angles with 360 degree and you will get the required value.
Let me show you the process with some examples;
Example 01
Find the value of missing angle x in below quadrilateral.
Solution
In the above quadrilateral;
∠A = 90
∠D = 90
∠B = 120
Applying angle sum property of quadrilateral;
∠A + ∠B + ∠C+∠D = 360
90 + 120 + ∠C + 90 = 360
300 + ∠C = 360
∠C = 360 – 300
∠C = 60 degree
Hence, value of x is 60 degree.
Example 02
In the below figure, find the measure of angle x.
Solution
In the above quadrilateral;
∠A = 120
∠D = 70
∠C = 65
Applying angle sum property of quadrilateral;
∠A + ∠B + ∠C+∠D = 360
120 + ∠B + 65 + 70 = 360
255 + ∠B = 360
∠B = 360 – 255
∠B = 105 degree
Hence, angle x measures 105 degree
Example 03
Given below is the parallelogram ABCD. Find the measure of all the angles.
Solution
In parallelogram, sum of adjacent angle measures 180 degrees.
∠B + ∠C = 180
x + 40 = 180
x = 140 degree
Also in parallelogram, opposite angles are equal in measure.
∠A = ∠C = 40 degree
∠D = ∠B = 140 degree
Example 04
Given below is the quadrilateral ABCD. Find the measure of all angles.
Solution
In quadrilateral, sum of all angles measures 360 degree.
∠A + ∠B + ∠C+∠D = 360
100 + (3x – 30) + (5x + 40) + 2x = 360
100 – 30 + 40 + (3x + 5x+ 2x) = 360
110 + 10x = 360
10x = 360 -110
10x = 250
x = 25
Now put the value of x in all the angles.
∠A = 100
∠B = 3x – 30 = 3(25) – 30 = 45
∠C = 5x + 40 = 5(25) + 40 = 165
∠D = 2x = 2 (25) = 50
Hence, we got the measure of all the angles.
Example 05
Given below is the quadrilateral ABCD. Find the measure of angle x.
Solution
First find interior ∠B of the given quadrilateral.
Since CB is a straight line, the sum of adjacent angle measure 180 degree.
∠B + 124 = 180
∠B = 180 – 124
∠B = 56 degree
Now apply angle sum property of quadrilateral.
∠A + ∠B + ∠C+∠D = 360
39 + 56 + 42 + x = 360
137 + x = 360
x = 360 – 137
x = 223
Hence, ∠x measures 223 degrees.