Find the perimeter of the trapezoid. Leave your answer in simples radical form

Perimeter = 105 + 15√3 in

Step-by-step explanation:

✔️Find the height of the trapezoid using trigonometric ratio:

Reference angle = 60°

opposite = height

Adj = 45 - 30 = 15 in

Thus:

Tan 60 = opp/adj

Tan 60 = height/15

height = 15*tan 60

height = 15√3 (tan 60 = √3)

✔️Find the slant height (hypotenuse) using Pythagorean Theorem:

Hypotenuse = √(height² + adj²)

Hypotenuse = √((15√3)² + 15²)

= √(675 + 225)

Hyp = √900

Hyp = 30

✔️Perimeter = 30 + 45 + 30 + 15√3

Perimeter = 105 + 15√3 in

m∠cdf = 43° , m∠gdf = 114°, m∠gde = 23°, m∠edh = 23° , m∠cdh = 157°

step-by-step explanation:

since, it is given that de bisects the angle gdh ⇒ de divides the angle gdh into two equal angles.

⇒ m∠gde = m∠edh

⇒ 8·x   - 1 = 6·x + 5

⇒ x = 3

so, m∠gde = 8·x - 1 = 23°

and m∠edh = 23°

now, it is given ∠cde is an straight angle so, m∠cde = 180°

⇒ m∠cdf + m∠fdg + m∠gde = 180°

⇒ 43° + m∠fdg + 23° = 180°

⇒ m∠fdg = 114°

again, m∠cde = m∠cdh + m∠edh = 180°

m∠cdh + 23° = 180°

⇒ m∠cdh = 157°