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.