### All Intermediate Geometry Resources

## Example Questions

### Example Question #1 : Rhombuses

Given: Rhombuses and .

True, false, or undetermined: Rhombus Rhombus .

**Possible Answers:**

Undetermined

False

True

**Correct answer:**

Undetermined

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 #1 : Rhombuses

Given: Rhombuses and .

and

True, false, or undetermined: Rhombus Rhombus .

**Possible Answers:**

Undetermined

False

True

**Correct answer:**

True

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?

**Possible Answers:**

Not enough information is provided to solve this problem.

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

Both of the remaining angles are degrees.

All of the interior angles of the rhombus are degrees.

**Correct answer:**

Both of the remaining angles are degrees.

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

**Possible Answers:**

**Correct answer:**

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 #1 : How To Find An Angle In A Rhombus

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

**Possible Answers:**

**Correct answer:**

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 #1 : How To Find An Angle In A Rhombus

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

**Possible Answers:**

**Correct answer:**

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 #1 : 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

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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 #1 : How To Find An Angle In A Rhombus

Given: Parallelogram such that .

True or false: Parallelogram cannot be a rhombus.

**Possible Answers:**

True

False

**Correct answer:**

False

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 #2 : How To Find An Angle In A Rhombus

Given: Rhombus with diagonals and intersecting at point .

True or false: must be a right angle.

**Possible Answers:**

False

True

**Correct answer:**

True

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