Question

1. find the area of the shaded portion of this trapezoid. dimensions are in feet.
2. find x and y.

Explanation:

Problem 29 (Area of shaded trapezoid portion)

Step1: Calculate trapezoid area

The area of a trapezoid is $\frac{1}{2} \times (base_1 + base_2) \times height$.
$\frac{1}{2} \times (11 + 17) \times 6 = \frac{1}{2} \times 28 \times 6 = 84$ square feet

Step2: Calculate unshaded triangle area

The area of a triangle is $\frac{1}{2} \times base \times height$.
$\frac{1}{2} \times 3 \times 4 = 6$ square feet

Step3: Subtract triangle from trapezoid

Subtract the unshaded area from the total trapezoid area to get the shaded area.
$84 - 6 = 78$

Step1: Identify equal sides property

The triangle has two equal sides, so it is isosceles, meaning the base angles are equal: $x^\circ$ is one base angle, and the other base angle is also $x^\circ$.

Step2: Use supplementary angle rule

The $140^\circ$ angle is supplementary to the adjacent interior angle of the triangle.
Interior angle = $180^\circ - 140^\circ = 40^\circ$

Step3: Solve for $x$ using triangle sum

The sum of angles in a triangle is $180^\circ$.
$2x + 40^\circ = 180^\circ$
$2x = 140^\circ$
$x = 70^\circ$

Step4: Solve for $y$ using exterior angle rule

The exterior angle $y^\circ$ equals the sum of the two non-adjacent interior angles.
$y = x + 40^\circ = 70^\circ + 40^\circ = 110^\circ$

Answer:

78 square feet

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Problem 30 (Find $x$ and $y$)