Question

Parallelogram's angles are in the ratio of 5:15:9:7.
What is the difference between largest and least angle of parallelogram?

Answer 1

In a parallelogram, opposite angles are equal and the sum of the angles is \(360^\circ\). Given that the angles are in the ratio of \(5:15:9:7\), we can represent the angles as:

- \(5x\)
- \(15x\)
- \(9x\)
- \(7x\)

Since opposite angles are equal, we can equate pairs of angles:

1. \(5x\) and \(15x\)
2. \(9x\) and \(7x\)

However, this representation is incorrect because we should consider that all four angles should only sum to \(360^\circ\). Thus, we can find the total ratio parts:

\[
5 + 15 + 9 + 7 = 36
\]

Let each angle be \(kx\), where \(k = 36\) and \(x\) is a common multiplier.

Then we can represent the angles as:

- \(5k = 5x\)
- \(15k = 15x\)
- \(9k = 9x\)
- \(7k = 7x\)

Setting the ratio to sum up to \(360^\circ\):

\[
(5 + 15 + 9 + 7)x = 360
\]

Calculating this gives:

\[
36x = 360
\]

Solving for \(x\):

\[
x = \frac{360}{36} = 10
\]

Now, we can find each angle:

- \(5x = 5 \cdot 10 = 50^\circ\)
- \(15x = 15 \cdot 10 = 150^\circ\)
- \(9x = 9 \cdot 10 = 90^\circ\)
- \(7x = 7 \cdot 10 = 70^\circ\)

The angles of the parallelogram are \(50^\circ\), \(150^\circ\), \(90^\circ\), and \(70^\circ\).

Now we need to identify the largest and smallest angles:

- Largest angle: \(150^\circ\)
- Smallest angle: \(50^\circ\)

Finally, the difference between the largest and smallest angle:

\[
150^\circ - 50^\circ = 100^\circ
\]

Thus, the difference between the largest and least angle of the parallelogram is \(100^\circ\).