How to find an angle in a parallelogram - ACT Math

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Question

In the parallellogram, what is the value of ?

Answer

Opposite angles are equal, and adjacent angles must sum to 180.

Therefore, we can set up an equation to solve for z:

(z – 15) + 2z = 180

3z - 15 = 180

3z = 195

z = 65

Now solve for x:

2_z_ = x = 130°

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Question

In parallelogram , $\measuredangle A = 45^{\circ}$ . What is $\measuredangle B?$

Answer

In the above parallelogram, $\measuredangle A$ and $\measuredangle B$ are consecutive angles (i.e. next to each other). In a parallelogram, consecutive angles are supplementary, meaning they add to $180^{\circ}$ .

$\measuredangle A + \measuredangle B = 180^{\circ}$

$45^{\circ} + \measuredangle B = 180^{\circ}$

$\measuredangle B = 135^{\circ}$

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Question

In parallelogram , $\measuredangle B = 117^{\circ}$ . What is $\measuredangle D$ ?

Answer

In parallelogram , $\measuredangle B$ and $\measuredangle D$ are opposite angles. In a parallelogram, opposite angles are congruent. This means these two angles are equal.

$\measuredangle D = \measuredangle B$

$\measuredangle D = 117^{\circ}$

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Question

In parallelogram , $\overline{AB} = 15$ and the height is . What is $\measuredangle A$ ?

Answer

We can start this problem by drawing the height and labeling the lengths with the given values.

When we do this, we can see that we have drawn a triangle inside the paralellogram including $\measuredangle A$ . Because we know the lengths of two sides of this triangle, we can use trigonometry to find $\measuredangle A$ .

With respect to $\measuredangle A$ , we know the values of the opposite and hypotenuse sides of the triangle. Thus, we can use the sine function to solve for $\measuredangle A$ .

$\sin \left (\measuredangle A\right ) = \frac{\textup{opposite}}{\textup{hypotenuse}}$

$\sin \left (\measuredangle A\right ) = \frac{12}{15}= 0.8$

$\measuredangle A = \arcsin\left ( 0.8\right ) = 53.1^{\circ}$

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Question

In parallelogram , what is the sum of $\measuredangle A$ and $\measuredangle D$ ?

Answer

In a parallelogram, consecutive angles are supplementary. $\measuredangle A$ and $\measuredangle D$ are consecutive, so their sum is $180^{\circ}$ .

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Question

In parallelogram , $\measuredangle B = \left (2x + 15 \right )^{\circ}$ and $\measuredangle C = x^{\circ}$ . Find .

Answer

In a parallelogram, consecutive angles are supplementary. Thus,

$\measuredangle B + \measuredangle C = 180^{\circ}$

$\left ( 2x + 15\right ) + x = 180$

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Question

is a parallelogram. Find .

Answer

In a parallelogram, consecutive angles are supplementary (i.e. add to $180^{\circ}$ ) and opposite angles are congruent (i.e. equal). Using these properties, we can write a system of equations.

1. $\measuredangle B = \measuredangle D$

2. $\measuredangle C + \measuredangle D = 180^{\circ}$

Starting with equation 1.,

Now substituting into equation 2.,

$y + \left ( y + x\right ) = 180$

Finally, because opposite angles are congruent, we know that $\measuredangle A = \measuredangle C$ .

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Question

is a parallelogram. Find .

Answer

In a parallelogram, consecutive angles are supplementary and opposite angles are congruent. Using these properties, we can write a system of equations. Because we have three variables, we will need three equations.

1. $\measuredangle A + \measuredangle B = 180^{\circ}$

2. $\measuredangle B = \measuredangle D$

3. $\measuredangle C + \measuredangle D = 180^{\circ}$

Start with equation 1.

$z + \left (x + z \right ) = 180$

Now simplify equation 2.

Finally, simplify equation 3.

$\left ( y + 20\right ) + \left ( y+ 2x\right ) = 180$

Note that we can plug this simplified equation 3 directly into the simplified equation 2 to solve for .

Now that we have , we can solve for using equation 1.

$x + 2\left ( 80\right ) = 180$

With , we can solve for using equation 3.

Now that we have and , we can solve for .

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Question

is a parallelogram. Find .

Answer

In a parallelogram, consecutive angles are supplementary and opposite angles are congruent. Using these properties, we can write a system of equations.

1. $\measuredangle B =\measuredangle D$

2. $\measuredangle B + \measuredangle C = 180^{\circ}$

3. $\measuredangle A = \measuredangle C$

Starting with equation 1.,

Substituting into equation 2.,

Using equation 3.,

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Question

In parallelogram , $\measuredangle C = 75^{\circ}$ . What is $\measuredangle D$ ?

Answer

In a parellelogram, consecutive angles are supplementary.

$\measuredangle C + \measuredangle D = 180^{\circ}$

$75^{\circ} + \measuredangle D = 180^{\circ}$

$\measuredangle D = 105^{\circ}$

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Question

In parallelogram , $\measuredangle D = 125^{\circ}$ . What is $\measuredangle B$ ?

Answer

In a parallelogram, opposite angles are congruent.

$\measuredangle B = \measuredangle D$

$\measuredangle B = 125^{\circ}$

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Question

is a parallelogram. Find .

Answer

In a parallelogram, consecutive angles are supplementary and opposite angles are congruent.

$\measuredangle A + \measuredangle D = 180^{\circ}$

$z + \left ( 2z + 9\right ) = 180$

$\measuredangle A = \measuredangle C$

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Question

is a parallelogram. Find .

Answer

In a parallelogram, consecutive angles are supplementary and opposite angles are congruent.

$\measuredangle A + \measuredangle D = 180^{\circ}$

$z + \left ( 2z + 9\right ) = 180$

$\measuredangle A + \measuredangle B = 180^{\circ}$

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