Question

directions: if each quadrilateral below is a trapezoid, find the missing measures.
1.
$m\angle c = \underline{\quad\quad\quad}$
$m\angle e = \underline{\quad\quad\quad}$
2.
$m\angle q = \underline{\quad\quad\quad}$
$m\angle s = \underline{\quad\quad\quad}$
3.
$m\angle j = \underline{\quad\quad\quad}$
$m\angle l = \underline{\quad\quad\quad}$
$m\angle m = \underline{\quad\quad\quad}$
4.
$m\angle w = \underline{\quad\quad\quad}$
$m\angle x = \underline{\quad\quad\quad}$
$m\angle z = \underline{\quad\quad\quad}$

1. solve for $x$.
2. find $m\angle b$.

7.
$m\angle m = \underline{\quad\quad\quad}$
$m\angle n = \underline{\quad\quad\quad}$
$m\angle o = \underline{\quad\quad\quad}$
$m\angle p = \underline{\quad\quad\quad}$
8.
$m\angle w = \underline{\quad\quad\quad}$
$m\angle x = \underline{\quad\quad\quad}$
$m\angle y = \underline{\quad\quad\quad}$
$m\angle z = \underline{\quad\quad\quad}$

Explanation:

Key Rule for Trapezoids:

In a trapezoid, consecutive angles between the two parallel bases are supplementary (sum to $180^\circ$). The total sum of interior angles of a quadrilateral is $360^\circ$.

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Problem 1

Step1: Find $m\angle C$

$\angle C$ and $\angle D$ are supplementary.
$m\angle C = 180^\circ - 134^\circ = 46^\circ$

Step2: Find $m\angle E$

$\angle E$ and $\angle F$ are supplementary.
$m\angle E = 180^\circ - 79^\circ = 101^\circ$

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Problem 2

Step1: Find $m\angle Q$

$\angle Q$ and $\angle R$ are supplementary.
$m\angle Q = 180^\circ - 27^\circ = 153^\circ$

Step2: Find $m\angle S$

$\angle S$ and $\angle T$ are supplementary.
$m\angle S = 180^\circ - 91^\circ = 89^\circ$

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Problem 3

This is an isosceles trapezoid (legs are congruent), so base angles are equal.

Step1: Find $m\angle J$

$\angle J$ and $\angle K$ are supplementary.
$m\angle J = 180^\circ - 83^\circ = 97^\circ$

Step2: Find $m\angle L$

Base angles are equal: $m\angle L = m\angle K = 83^\circ$

Step3: Find $m\angle M$

Base angles are equal: $m\angle M = m\angle J = 97^\circ$

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Problem 4

This is an isosceles trapezoid (legs are congruent).

Step1: Find $m\angle Z$

$\angle Z$ and $\angle Y$ are supplementary.
$m\angle Z = 180^\circ - 146^\circ = 34^\circ$

Step2: Find $m\angle X$

Base angles are equal: $m\angle X = m\angle Y = 146^\circ$

Step3: Find $m\angle W$

Base angles are equal: $m\angle W = m\angle Z = 34^\circ$

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Problem 5

$\angle S$ and $\angle V$ are supplementary.

Step1: Set up supplementary equation

$14x - 15 + 139 = 180$

Step2: Simplify and solve for $x$

$14x + 124 = 180$
$14x = 180 - 124 = 56$
$x = \frac{56}{14} = 4$

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Problem 6

$\angle B$ and $\angle C$ are supplementary.

Step1: Set up supplementary equation

$9x + 2 + 5x - 4 = 180$

Step2: Simplify and solve for $x$

$14x - 2 = 180$
$14x = 182$
$x = \frac{182}{14} = 13$

Step3: Calculate $m\angle B$

$m\angle B = 9(13) + 2 = 117 + 2 = 119^\circ$

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Problem 7

This is an isosceles trapezoid, so $\angle M = \angle N$.

Step1: Set equal angle equation

$8x - 16 = 6x + 20$

Step2: Solve for $x$

$8x - 6x = 20 + 16$
$2x = 36$
$x = 18$

Step3: Find $m\angle M$ and $m\angle N$

$m\angle M = 8(18) - 16 = 144 - 16 = 128^\circ$
$m\angle N = m\angle M = 128^\circ$

Step4: Find $m\angle O$ and $m\angle P$

$\angle O$ and $\angle N$ are supplementary: $m\angle O = 180^\circ - 128^\circ = 52^\circ$
$m\angle P = m\angle O = 52^\circ$

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Problem 8

This is an isosceles trapezoid, so $\angle X = \angle Y$.

Step1: Set equal angle equation

$13x - 7 = 8x - 2$

Step2: Solve for $x$

$13x - 8x = -2 + 7$
$5x = 5$
$x = 1$

Step3: Find $m\angle X$ and $m\angle Y$

$m\angle X = 13(1) - 7 = 6^\circ$
$m\angle Y = m\angle X = 6^\circ$

Step4: Find $m\angle W$ and $m\angle Z$

$\angle W$ and $\angle X$ are supplementary: $m\angle W = 180^\circ - 6^\circ = 174^\circ$
$m\angle Z = m\angle W = 174^\circ$

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Answer:

1. $m\angle C = 46^\circ$, $m\angle E = 101^\circ$
2. $m\angle Q = 153^\circ$, $m\angle S = 89^\circ$
3. $m\angle J = 97^\circ$, $m\angle L = 83^\circ$, $m\angle M = 97^\circ$
4. $m\angle W = 34^\circ$, $m\angle X = 146^\circ$, $m\angle Z = 34^\circ$
5. $x = 4$
6. $m\angle B = 119^\circ$
7. $m\angle M = 128^\circ$, $m\angle N = 128^\circ$, $m\angle O = 52^\circ$, $m\angle P = 52^\circ$
8. $m\angle W = 174^\circ$, $m\angle X = 6^\circ$, $m\angle Y = 6^\circ$, $m\angle Z = 174^\circ$