Question

1. if pqrs is a parallelogram, find the values of x and y.11. x = _____y = ___12. if abcd is a parallelogram, find m∠c.12. m∠c = _____determine whether the quadrilateral is a parallelogram using the indicated method.13. d(-8, 1), e(-3, 6), f(7, 4), g(2, -1) (distance formula)13. yes / no14. l(-1, 6), m(5, 9), n(0, 2), p(-8, -2) (slope formula)14. yes / no15. b(-2, -9), c(0, -5), d(6, -3), e(4, -7) (distance and slope formulas)15. yes / no

Explanation:

Problem 11:

Step1: Set equal opposite sides

$13x + 15 = 19x - 9$

Step2: Solve for x

$15 + 9 = 19x - 13x$
$24 = 6x$
$x = \frac{24}{6} = 4$

Step3: Set equal opposite angles

$4y + 7 = 10y - 32$

Step4: Solve for y

$7 + 32 = 10y - 4y$
$39 = 6y$
$y = \frac{39}{6} = 6.5$

Problem 12:

Step1: Set consecutive angles supplementary

$(5x + 38) + (8x - 19) = 180$

Step2: Simplify and solve for x

$13x + 19 = 180$
$13x = 161$
$x = \frac{161}{13} = 12.38$ (rounded)

Step3: Find $m\angle C$ (equal to $\angle A$; $\angle A$ supplementary to $\angle B$)

$m\angle C = 180 - (5(12.38) + 38)$
$m\angle C = 180 - (61.9 + 38) = 180 - 99.9 = 80.1^\circ$
Or use $\angle C = \angle A = 8x - 19 = 8(12.38)-19=99.04-19=80.04^\circ \approx 80^\circ$

Problem 13:

Step1: Calculate all side distances

Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$DE=\sqrt{(-3+8)^2+(6-1)^2}=\sqrt{25+25}=\sqrt{50}$
$FG=\sqrt{(2-7)^2+(-1-4)^2}=\sqrt{25+25}=\sqrt{50}$
$EF=\sqrt{(7+3)^2+(4-6)^2}=\sqrt{100+4}=\sqrt{104}$
$GD=\sqrt{(-8-2)^2+(1+1)^2}=\sqrt{100+4}=\sqrt{104}$

Step2: Check opposite sides equal

Opposite sides are equal, so it is a parallelogram.

Problem 14:

Step1: Calculate all side slopes

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_{LM}=\frac{9-6}{5+1}=\frac{3}{6}=\frac{1}{2}$
$m_{NP}=\frac{-2-2}{-8-0}=\frac{-4}{-8}=\frac{1}{2}$
$m_{MN}=\frac{2-9}{0-5}=\frac{-7}{-5}=\frac{7}{5}$
$m_{PL}=\frac{6+2}{-1+8}=\frac{8}{7}$

Step2: Check opposite sides parallel

$m_{MN}
eq m_{PL}$, so opposite sides not all parallel.

Problem 15:

Step1: Check slopes of opposite sides

$m_{BC}=\frac{-5+9}{0+2}=\frac{4}{2}=2$
$m_{ED}=\frac{-3+7}{6-4}=\frac{4}{2}=2$
$m_{CD}=\frac{-3+5}{6-0}=\frac{2}{6}=\frac{1}{3}$
$m_{BE}=\frac{-7+9}{4+2}=\frac{2}{6}=\frac{1}{3}$

Step2: Check distances of opposite sides

$BC=\sqrt{(0+2)^2+(-5+9)^2}=\sqrt{4+16}=\sqrt{20}$
$ED=\sqrt{(6-4)^2+(-3+7)^2}=\sqrt{4+16}=\sqrt{20}$
$CD=\sqrt{(6-0)^2+(-3+5)^2}=\sqrt{36+4}=\sqrt{40}$
$BE=\sqrt{(4+2)^2+(-7+9)^2}=\sqrt{36+4}=\sqrt{40}$

Step3: Verify parallel and equal sides

Opposite sides are parallel and equal, so it is a parallelogram.

Answer:

1. $x=4$, $y=6.5$
2. $m\angle C \approx 80^\circ$
3. YES
4. NO
5. YES