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:
- \(5x\) and \(15x\)
- \(9x\) 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 \(100^\circ\).
2 answers