# Intermediate Geometry : Rhombuses

## Example Questions

← Previous 1 3 4 5 6 7 8 9 10

### Example Question #1 : Rhombuses

Given: Rhombuses  and .

True, false, or undetermined: Rhombus  Rhombus .

True

Undetermined

False

Undetermined

Explanation:

Two figures are congruent by definition if all of their corresponding sides are congruent and all of their corresponding angles are congruent.

By definition, a rhombus has four sides of equal length.  If we let  be the common sidelength of Rhombus  and  be the common sidelength of Rhombus , then, since , it follows that , so corresponding sides are congruent. However, no information is given about their angle measures. Therefore, it cannot be determined whether or not the two rhombuses are congruent.

### Example Question #2 : Rhombuses

Given: Rhombuses  and .

and

True, false, or undetermined: Rhombus  Rhombus .

False

True

Undetermined

True

Explanation:

Two figures are similar by definition if all of their corresponding sides are proportional and all of their corresponding angles are congruent.

By definition, a rhombus has four sides that are congruent. If we let  be the common sidelength of Rhombus  and  be the common sidelength of Rhombus , it can easily be seen that the ratio of the length of each side of the former to that of the latter is the same ratio, namely, .

Also, a rhombus being a parallelogram, its opposite angles are congruent, and its consecutive angles are supplementary. Therefore, since , it follows that , and . By a similar argument,  and . Therefore,

Since all corresponding sides are proportional and all corresponding angles are congruent, it holds that Rhombus  Rhombus .

### Example Question #1 : How To Find An Angle In A Rhombus

A rhombus has two interior angles with a measurement of  degrees. What is the measurement of each of the other two interior angles?

Both of the remaining angles are  degrees.

One angle is  degrees and the other angle must be  degrees.

All of the interior angles of the rhombus are  degrees.

Not enough information is provided to solve this problem.

Both of the remaining angles are  degrees.

Explanation:

The four interior angles in any rhombus must have a sum of  degrees. The opposite interior angles must be equivalent, and the adjacent angles have a sum of  degrees.

Thus, if a rhombus has two interior angles of  degrees, there must also be two angles that equal:

Check:

### Example Question #2 : How To Find An Angle In A Rhombus

Using the rhombus above, find the sum of angles  and

Explanation:

The four interior angles in any rhombus must have a sum of  degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of  degrees.

Since  then,

### Example Question #3 : How To Find An Angle In A Rhombus

In the above rhombus, angle  degrees. Find the sum of angles  and

Explanation:

The four interior angles in any rhombus must have a sum of  degrees. The opposite interior angles must be equivalent, and the adjacent angles have a sum of  degrees.

Since ,

And

So,

### Example Question #4 : How To Find An Angle In A Rhombus

In the rhombus shown above, angle  has a measurement of  degrees. Find the measurement of angle

Explanation:

The four interior angles in any rhombus must have a sum of  degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of  degrees.

Since angle  is adjacent to angle , they must have a sum of  degrees.

The solution is:

### Example Question #5 : How To Find An Angle In A Rhombus

In the rhombus shown above, angle  has a measurement of  degrees. Find the sum of angles  and

Explanation:

The four interior angles in any rhombus must have a sum of  degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of  degrees.

Angles  and  are opposite interior angles, so they must have equivalent measurements.

The sum is:

### Example Question #6 : How To Find An Angle In A Rhombus

Angle  has a measurement of  degrees. Find the sum of angles  and

Explanation:

The four interior angles in any rhombus must have a sum of  degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of  degrees.

Since, both angles  and  are adjacent to angle --find the measurement of one of these two angles by: .

Angle  and angle  must each equal  degrees. So the sum of angles  and  degrees.

### Example Question #7 : How To Find An Angle In A Rhombus

Given: Parallelogram  such that .

True or false: Parallelogram  cannot be a rhombus.

True

False

False

Explanation:

A rhombus is defined to be a parallelogram with four congruent sides; there is no restriction as to the measures of the angles. Therefore, a rhombus can have angles of any measure. The correct choice is "false".

### Example Question #1 : How To Find An Angle In A Rhombus

Given: Rhombus  with diagonals  and  intersecting at point .

True or false:  must be a right angle.

False

True

True

Explanation:

One characteristic of a rhombus is that its diagonals are perpendicular. It follows that  must be a right angle.

← Previous 1 3 4 5 6 7 8 9 10