In a trapezium ABCD , angleA :angleD = 3:2 and angle B : angle C = 4:5 . Find all angles of trapezium .
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To find all the angles of a trapezium ABCD, we need to use the given ratios of angles A, B, C, and D.
Let's start by assigning variables to the angles. Let angle A be 3x, angle D be 2x, angle B be 4y, and angle C be 5y.
Since the sum of the angles in a trapezium is 360 degrees, we can write the equation:
3x + 2x + 4y + 5y = 360
Simplifying the equation, we get:
5x + 9y = 360 (equation 1)
Now, we need to use the given ratios to set up equations involving x and y.
We know that angle A : angle D = 3 : 2. So, we can write the equation:
(3x) : (2x) = 3 : 2
Cross-multiplying, we have:
3(2x) = 2(3x)
6x = 6x
This equation shows that angle A is equal to angle D, indicating that they are congruent.
Similarly, we have the ratio of angle B : angle C = 4 : 5. Writing this as an equation, we get:
(4y) : (5y) = 4 : 5
Cross-multiplying, we have:
4(5y) = 5(4y)
20y = 20y
This equation shows that angle B is equal to angle C, indicating that they are congruent.
Now, let's go back to equation 1 and substitute the congruent angles:
5x + 9y = 360
Since angle A = angle D, we can write it as:
5x + 9y = 360 (equation 2)
We can simplify equation 2 by dividing both sides by 5:
x + (9/5)y = 72
Now, we need to choose a value for y that makes the equation solvable. Let's assume y = 5.
Substituting y = 5 into the equation, we have:
x + (9/5)(5) = 72
x + 9 = 72
Subtracting 9 from both sides:
x = 63
Therefore, x = 63 and y = 5.
Now, let's find the angles of the trapezium by substituting the values of x and y into our variable assignments:
Angle A = 3x = 3(63) = 189 degrees
Angle B = 4y = 4(5) = 20 degrees
Angle C = 5y = 5(5) = 25 degrees
Angle D = 2x = 2(63) = 126 degrees
So, the angles of the trapezium ABCD are: A = 189 degrees, B = 20 degrees, C = 25 degrees, and D = 126 degrees.