Question

coloring activity!date: _____________ per: __directions: a-c are parallelograms, d-f are rectangles, g-j are rhombi, and k-l are squares. find each part, identify the answers from the back, and record the number with the color. color the picture!a find $m\angle b$ and $m\angle c$.$m\angle b =$ number$m\angle c =$ colorb find $m\angle p$ and $m\angle q$.$m\angle p =$ number$m\angle q =$ colorc if $xv = 27 - 2x$ and $vz = 3x + 2$, find $vz$ and $xz$.$vz =$ number$xz =$ colord find $m\angle egf$ and $m\angle dge$.$m\angle egf =$ number$m\angle dge =$ colore if $jl = 5x + 1$ and $mk = 8x - 32$, find $jl$ and $nk$.$jl =$ number$nk =$ colorf if $rv = x + 6$, $us = 5x - 9$, and $rs = 11$, find $vt$ and $st$.$vt =$ number$st =$ color© gina wilson (all things algebra), 2016

Explanation:

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Part A

Step1: Set supplementary angles equal to 180°

In a parallelogram, consecutive angles are supplementary:
$$(10x - 19) + (7x + 23) = 180$$

Step2: Solve for $x$

Simplify and isolate $x$:

$$17x + 4 = 180 \\ 17x = 176 \\ x = 10$$

Step3: Calculate $m\angle B$

Substitute $x=10$ into $\angle B$:
$$m\angle B = 10(10) - 19 = 81^\circ$$

Step4: Calculate $m\angle C$

In a parallelogram, opposite angles are equal:
$$m\angle C = m\angle D = 7(10) + 23 = 93^\circ$$

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Part B

Step1: Set supplementary angles equal to 180°

In a parallelogram, consecutive angles are supplementary:
$$(5x + 3) + (14x + 6) = 180$$

Step2: Solve for $x$

Simplify and isolate $x$:

$$19x + 9 = 180 \\ 19x = 171 \\ x = 9$$

Step3: Calculate $m\angle Q$

Substitute $x=9$ into $\angle Q$ (opposite $\angle R$):
$$m\angle Q = m\angle R = 5(9) + 3 = 48^\circ$$

Step4: Calculate $m\angle P$

In a parallelogram, opposite angles are equal:
$$m\angle P = m\angle S = 14(9) + 6 = 132^\circ$$

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Part C

Step1: Set diagonals' segments equal

In a parallelogram, diagonals bisect each other:
$$27 - 2x = 3x + 2$$

Step2: Solve for $x$

Simplify and isolate $x$:

$$25 = 5x \\ x = 5$$

Step3: Calculate $VZ$

Substitute $x=5$ into $VZ$:
$$VZ = 3(5) + 2 = 17$$

Step4: Calculate $XZ$

$XZ = XV + VZ = 2 \times VZ$:
$$XZ = 2(17) = 34$$

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Part D

Step1: Set alternate interior angles equal

In a rectangle, $DE \parallel GF$, so alternate interior angles are equal:
$$11x - 10 = 2x + 17$$

Step2: Solve for $x$

Simplify and isolate $x$:

$$9x = 27 \\ x = 3$$

Step3: Calculate $m\angle EGF$

Substitute $x=3$ into $\angle EGF$:
$$m\angle EGF = 2(3) + 17 = 23^\circ$$

Step4: Calculate $m\angle DGE$

$\angle DGE$ is complementary to $\angle EGF$ (rectangle corner is 90°):
$$m\angle DGE = 90^\circ - 23^\circ = 67^\circ$$

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Part E

Step1: Set diagonals equal

In a rectangle, diagonals are congruent:
$$5x + 1 = 8x - 32$$

Step2: Solve for $x$

Simplify and isolate $x$:

$$33 = 3x \\ x = 11$$

Step3: Calculate $JL$

Substitute $x=11$ into $JL$:
$$JL = 5(11) + 1 = 56$$

Step4: Calculate $NK$

In a rectangle, diagonals bisect each other, so $NK = \frac{1}{2}MK = \frac{1}{2}JL$:
$$NK = \frac{56}{2} = 28$$

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Part F

Step1: Set diagonals' segments equal

In a rectangle, diagonals bisect each other:
$$x + 6 = 5x - 9$$

Step2: Solve for $x$

Simplify and isolate $x$:

$$15 = 4x \\ x = 3.75$$

Step3: Calculate $VT$

Diagonals are equal to side $RS=11$:
$$VT = 11$$

Step4: Calculate $ST$

$ST = RS = 11$ (sides of rectangle are equal to the given side length)
$$ST = 11$$

Answer:

Part A

$m\angle B = 81^\circ$
$m\angle C = 93^\circ$

Part B

$m\angle P = 132^\circ$
$m\angle Q = 48^\circ$

Part C

$VZ = 17$
$XZ = 34$

Part D

$m\angle EGF = 23^\circ$
$m\angle DGE = 67^\circ$

Part E

$JL = 56$
$NK = 28$

Part F

$VT = 11$
$ST = 11$