Question

1. (\triangle def) is the image of (\triangle def) after a dilation with a scale factor of 2. what are the coordinates of the vertices of (\triangle def)?
2. consider pentagon (abcde).

part a: find the value of (x).
part b: determine (mangle b), (mangle c), and (mangle e).
part c: find the value of each exterior angle.

Explanation:

Step1: Use the formula for the sum of interior angles of a pentagon

The sum of interior angles of a pentagon is \((5 - 2)\times180^{\circ}=540^{\circ}\).
So, \(124+(19x)+113+(98 + 3x)+(100 - x)=540\).

Step2: Combine like terms

Combine the constant terms: \(124+113+98+100 = 435\).
Combine the \(x\) - terms: \(19x+3x - x=21x\).
The equation becomes \(435+21x = 540\).

Step3: Solve for \(x\)

Subtract \(435\) from both sides: \(21x=540 - 435\), so \(21x = 105\).
Divide both sides by \(21\): \(x=\frac{105}{21}=5\).

Step4: Find \(m\angle B\), \(m\angle C\) and \(m\angle E\)

For \(m\angle B\): Substitute \(x = 5\) into \(98 + 3x\).
\(m\angle B=98+3\times5=98 + 15=113^{\circ}\).
For \(m\angle C\): Substitute \(x = 5\) into \(100 - x\).
\(m\angle C=100-5 = 95^{\circ}\).
For \(m\angle E\): Substitute \(x = 5\) into \(19x\).
\(m\angle E=19\times5=95^{\circ}\).

Step5: Find the exterior angles

The exterior angle of an interior angle \(A\) is \(180 - A\).
For \(\angle A = 113^{\circ}\), exterior angle \(=180-113 = 67^{\circ}\).
For \(\angle B = 113^{\circ}\), exterior angle \(=180 - 113=67^{\circ}\).
For \(\angle C = 95^{\circ}\), exterior angle \(=180-95 = 85^{\circ}\).
For \(\angle D = 124^{\circ}\), exterior angle \(=180-124 = 56^{\circ}\).
For \(\angle E = 95^{\circ}\), exterior angle \(=180-95=85^{\circ}\).

Answer:

Part A: \(x = 5\).
Part B: \(m\angle B=113^{\circ}\), \(m\angle C = 95^{\circ}\), \(m\angle E=95^{\circ}\).
Part C: The exterior angles are \(67^{\circ}\), \(67^{\circ}\), \(85^{\circ}\), \(56^{\circ}\), \(85^{\circ}\).