### All Intermediate Geometry Resources

## Example Questions

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

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

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

Not enough information is provided to find an answer.

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

True

False

**Correct answer:**

False

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.

**Possible Answers:**

False

True

**Correct answer:**

False

, 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.

**Possible Answers:**

True

False

**Correct answer:**

True

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.

**Possible Answers:**

True

False

**Correct answer:**

False

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.

**Possible Answers:**

False

True

**Correct answer:**

True

A rectangle is a parallelogram with four right angles.

Consecutive angles of a parallelogram are supplementary. If one angle of a parallelogram is given to be right, then its neighboring angles, being supplementary to a right angle, are right as well; also, opposite angles of a parallelogram are congruent, so the opposite angle is also right. All four angles must be right, making the parallelogram a rectangle by definition.