Question

1. find the value of x in the regular pentagon. (example 3) 9. find the measure of each angle of the quadrilateral. (example 4)

Explanation:

Question 7

Step1: Find the sum of interior angles of a pentagon

The formula for the sum of interior angles of a polygon is \((n - 2)\times180^{\circ}\), where \(n\) is the number of sides. For a pentagon, \(n = 5\). So the sum is \((5 - 2)\times180^{\circ}=540^{\circ}\).

Step2: Set up the equation for the regular pentagon

Since it is a regular pentagon, all interior angles are equal. Let the interior angle be \(\frac{1}{2}x + 50\). Then \(5\times(\frac{1}{2}x + 50)=540\).

Step3: Solve the equation

First, expand: \(\frac{5}{2}x+250 = 540\).
Subtract \(250\) from both sides: \(\frac{5}{2}x=540 - 250=290\).
Multiply both sides by \(\frac{2}{5}\): \(x=\frac{290\times2}{5}=116\).

Step1: Find the sum of interior angles of a quadrilateral

The formula for the sum of interior angles of a polygon is \((n - 2)\times180^{\circ}\), where \(n\) is the number of sides. For a quadrilateral, \(n = 4\). So the sum is \((4 - 2)\times180^{\circ}=360^{\circ}\).

Step2: Set up the equation

We have \(x+2x + 3x+4x=360\).
Combine like - terms: \((1 + 2+3 + 4)x=360\), i.e., \(10x=360\).

Step3: Solve for \(x\)

Divide both sides by \(10\): \(x = 36\).

Step4: Find the measure of each angle

\(\angle W=x^{\circ}=36^{\circ}\), \(\angle X = 2x^{\circ}=2\times36^{\circ}=72^{\circ}\), \(\angle Z=3x^{\circ}=3\times36^{\circ}=108^{\circ}\), \(\angle Y=4x^{\circ}=4\times36^{\circ}=144^{\circ}\).

Answer:

\(x = 116\)

Question 9