Question

for what value of x is the quadrilateral a parallelogram?
5.
diagram of a quadrilateral with diagonals labeled 2x - 1 and x + 6
6.
diagram of a quadrilateral with segments labeled 3x - 11 and x + 5
7.
diagram of a quadrilateral with diagonals labeled 8x and 3x + 5
8.
diagram of a quadrilateral with angles labeled 101°, x°, and 101°
9.
diagram of a quadrilateral with angles labeled 4x°, 5x°, and 4x°
10.
diagram of a parallelogram with angles labeled (5x - 7)° and (5x + 1)°

Explanation:

Step1: Set equal diagonals (Q6 top-left)

$2x - 1 = x + 6$

Step2: Solve for x

$2x - x = 6 + 1 \implies x=7$

Step1: Set equal segments (Q6 top-right)

$3x - 11 = x + 5$

Step2: Solve for x

$3x - x = 5 + 11 \implies 2x=16 \implies x=8$

Step1: Set equal diagonals (Q7)

$8x = 3x + 5$

Step2: Solve for x

$8x - 3x = 5 \implies 5x=5 \implies x=1$

Step1: Use consecutive angles sum (Q8)

$x + 101 = 180$

Step2: Solve for x

$x=180-101 \implies x=79$

Step1: Sum of quadrilateral angles

$4x + 5x + 4x + 5x = 360$

Step2: Simplify and solve

$18x=360 \implies x=\frac{360}{18}=36$

Step1: Set angles supplementary (Q10)

$5x - 7 + x + 1 = 180$

Step2: Simplify and solve

$6x -6=180 \implies 6x=186 \implies x=31$
*Correction: Use alternate interior angles (parallel sides):
$5x -7 = x +17$
$5x -x=17+7 \implies 4x=24 \implies x=6$*

Answer:

1. $x=7$
2. $x=8$
3. $x=1$
4. $x=79$
5. $x=36$
6. $x=6$