ACT Math Help » Geometry » Plane Geometry » Quadrilaterals » Trapezoids » How to find an angle in a trapezoid

Find the measure of angle 

*
in the isosceles trapezoid pictured below.

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*


*


*


*


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°.


Explanation:

As a rule, adjacent (non-paired) angles in a trapezoid are supplementary. Thus, we know that if 

*
, then 
*
. Since we are told that 
*
 and 
*
 are paired and trapezoid 
*
 is isosceles, 
*
 must also equal 
*
.


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,

*
 and
*
 are equal to each other. The two diagonals within the trapezoid bisect angles
*
 and 
*
 at the same angle.

Thus,

*
 must also be equal to 50 degrees.

Thus,

*
.

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

*
:

*


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:

*


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