ACT Math - How to find an angle in a trapezoid

Given the following isosceles triangle:

In degrees, find the measure of the sum of $\angle x$ and $\angle$ in the figure above.

$120^\circ$

$30^\circ$

$60^\circ$

$180^\circ$

$240^\circ$

Explanation

All quadrilaterals' interior angles sum to 360°. In isosceles trapezoids, the two top angles are equal to each other.

Similarly, the two bottom angles are equal to each other as well.

Therefore, to find the sum of the two bottom angles, we subtract the measures of the top two angles from 360:

In the isosceles trapezoid above,

$\overline{AB}\left | \right | \overline{DC}$ . $\angle BDC = 50^{\circ}$ and $\angle BCA = 15^{\circ}$ .

In degrees, what is the measure of $\angle DAC$ ?

$65^\circ$

$115^\circ$

$75^\circ$

$35^\circ$

$80^\circ$

Explanation

To find the measure of angle DAC, we must know that the interior angles of all triangles sum up to 180 degrees. Also, as this is an isosceles trapezoid, $\angle ADC$ and $\angle BCD$ are equal to each other. The two diagonals within the trapezoid bisect angles $\angle ADC$ and $\angle BCD$ at the same angle.

Thus, $\angle ACD$ must also be equal to 50 degrees.

Thus, $\angle ADC = 50 + 15 = 65$ .

Now that we know two angles out of the three in the triangle on the left, we can subtract them from 180 degrees to find $\angle DAC$ :

Find the measure of angle in the isosceles trapezoid pictured below.

Screen_shot_2013-03-18_at_3.32.40_pm

$18^{\circ}$

$36^{\circ}$

$72^{\circ}$

$108^{\circ}$

$118^{\circ}$

Explanation

The sum of the angles in any quadrilateral is 360**°, and the properties of an isosceles trapezoid dictate that the sets of angles adjoined by parallel lines (in this case, the bottom set and top set of angles) are equal. Subtracting 2(72°) from 360°** gives the sum of the two top angles, and dividing the resulting 216**°** by 2 yields the measurement of x, which is 108**°**.

Trapezoid is an isosceles trapezoid with angle $\angle C = 35^{\circ}$ . If $\angle A$ and $\angle B$ are paired, what is the measure of $\angle A$ ?

$145^{\circ}$

$30^{\circ}$

$35^{\circ}$

$55^{\circ}$

$215^{\circ}$

Explanation

As a rule, adjacent (non-paired) angles in a trapezoid are supplementary. Thus, we know that if $\angle C = 35^{\circ}$ , then $\angle B = 145^{\circ}$ . Since we are told that $\angle A$ and $\angle B$ are paired and trapezoid is isosceles, $\angle A$ must also equal $145^{\circ}$ .

How to find an angle in a trapezoid

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