Question

find the values of the variables or missing values in each trapezoid.
find the values of the variables for the given quadrilaterals.
find the values of the variables in each parallelogram.

Explanation:

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Trapezoid 1 (TWR S):

Step1: Use isosceles trapezoid angle property

Base angles are equal, so $\angle S = \angle R$.
$60 = 5x$

Step2: Solve for $x$

$x = \frac{60}{5} = 12$

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Trapezoid 2 (EFG H):

Step1: Use midsegment theorem

Midsegment = $\frac{1}{2}\times$ (sum of bases)
$2x + 4 = \frac{1}{2}(x + 4x + 7)$

Step2: Simplify and solve for $x$

$2(2x + 4) = 5x + 7$
$4x + 8 = 5x + 7$
$x = 8 - 7 = 1$

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Isosceles Trapezoid:

Step1: Legs of isosceles trapezoid are equal

$4 = 2x + 2$

Step2: Solve for $x$

$2x = 4 - 2 = 2$
$x = 1$

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Rhombus:

Step1: All sides of rhombus are equal

$5x = 15$

Step2: Solve for $x$

$x = \frac{15}{5} = 3$

Step3: Solve for $y$

$3y = 15$
$y = \frac{15}{3} = 5$

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Kite:

Step1: Pair of adjacent sides equal

$x = 3x - 4$ and $y + 1 = 2y - 5$

Step2: Solve for $x$

$3x - x = 4$
$2x = 4$
$x = 2$

Step3: Solve for $y$

$2y - y = 1 + 5$
$y = 6$

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Square:

Step1: All sides of square are equal

$2x - 7 = y - 1$ and $y - 1 = 2y - 5$

Step2: Solve for $y$

$2y - y = 5 - 1$
$y = 4$

Step3: Substitute $y$ to find $x$

$2x - 7 = 4 - 1 = 3$
$2x = 3 + 7 = 10$
$x = 5$

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Parallelogram 1:

Step1: Consecutive angles are supplementary

$6a + 10 + 130 = 180$

Step2: Solve for $a$

$6a = 180 - 140 = 40$
$a = \frac{40}{6} = \frac{20}{3} \approx 6.67$

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Parallelogram 2:

Step1: Opposite sides of parallelogram are equal

$7x - 1 = 6x + 2$ and $2x + 4 = 10$

Step2: Solve for $x$ (from right side)

$2x = 10 - 4 = 6$
$x = 3$
(Verify: $7(3)-1=20$, $6(3)+2=20$, which matches)

Answer:

1. Trapezoid (TWR S): $x = 12$
2. Trapezoid (EFG H): $x = 1$
3. Isosceles Trapezoid: $x = 1$
4. Rhombus: $x = 3$, $y = 5$
5. Kite: $x = 2$, $y = 6$
6. Square: $x = 5$, $y = 4$
7. Parallelogram 1: $a = \frac{20}{3}$
8. Parallelogram 2: $x = 3$