Intermediate Geometry : How to find an angle in a parallelogram

Example Questions

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Example Question #11 : How To Find An Angle In A Parallelogram

In the parallelogram shown above, angle  is  degrees. Find the measure of angle

Explanation:

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal  degrees. And, the adjacent interior angles must be supplementary angles (sum of  degrees).

Since, angles  and  are opposite interior angles, thus they must be equivalent.

, therefore

Example Question #12 : How To Find An Angle In A Parallelogram

In the parallelogram shown above, angle  is  degrees. Find the sum of angles  and

Explanation:

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal  degrees. And, the adjacent interior angles must be supplementary angles (sum of  degrees).

Thus, the solution is:

Since both angles  and  equal  There sum must equal

Example Question #13 : How To Find An Angle In A Parallelogram

Using the parallelogram above, find the sum of angles  and

Not enough information is provided to find an answer.

Explanation:

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal  degrees.

Also, the adjacent interior angles must be supplementary angles (sum of  degrees).

Since, angles  and  are adjacent to each other they must be supplementary angles.

Thus, the sum of these two angles must equal  degrees.

Example Question #14 : How To Find An Angle In A Parallelogram

A paralellogram as two angles that are 65 degrees and 115 degrees respectively.  What are the other two angles in the paralellogram?

Explanation:

This question is very simple to answer if you remember that ALL paralellograms have two pairs of equal and opposite angles, and that the four angles in any quadrilateral MUST add up to 360 degrees.

Because the angles given are different, we know that they are supplementary and the other two missing angles MUST be the same.

Example Question #15 : How To Find An Angle In A Parallelogram

Given: Regular Pentagon  with center . Construct segments  and  to form Quadrilateral .

True or false: Quadrilateral  is a parallelogram.

True

False

False

Explanation:

Below is regular Pentagon  with center , a segment drawn from  to each vertex - that is, each of its radii drawn.

The measure of each angle of a regular pentagon can be calculated by setting  equal to 5 in the formula

and evaluating:

Specifically,

By symmetry, each radius bisects one of these angles. Specifically,

By the Same-Side Interior Angles Theorem, consecutive angles of a parallelogram are supplementary - that is, their measures total . However,

,

violating these conditions. Therefore, Quadrilateral  is not a parallelogram.

Example Question #16 : How To Find An Angle In A Parallelogram

Given: Quadrilateral  such that  and .

True or false: It follows that Quadrilateral  is a parallelogram.

False

True

False

Explanation:

, making  and  supplementary. By the Converse of the Same Side Interior Angles Theorem, , it does follow that . However, without knowing the measures of the other two angles, nothing further can be concluded about Quadrilateral . Below are a parallelogram and a trapezoid, both of which have these two angles of these measures.

Example Question #17 : How To Find An Angle In A Parallelogram

Given: Parallelogram such that and .

True or false: It follows that Parallelogram is a rectangle.

True

False

True

Explanation:

By the Same-Side Interior Angles Theorem, consecutive angles of a parallelogram can be proved to be supplementary - that is, their angle measures total . Specifically,  and are a pair of supplementary angles. Since they are also congruent, it follows that both are right angles. For the same reason,  and  are also right angles. The parallelogram, having four right angles, is a rectangle by definition.

Example Question #18 : How To Find An Angle In A Parallelogram

Given: Rectangle  with diagonals  and  intersecting at point .

True or false:  must be a right angle.

True

False

False

Explanation:

The diagonals of a parallelogram are perpendicular - and, consequently,  is a right angle. - if and only if the parallelogram is a rhombus, a figure with four sides of equal length. Not all rectangles have four congruent sides.  Therefore,  need not be a right angle.

Example Question #19 : How To Find An Angle In A Parallelogram

Given: Parallelogram  such that .

True or false: Parallelogram  must be a rectangle.

False

True