Practice Problems
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1.
Which quadrilateral has two pairs of parallel sides and does not have a right angle?
 a. parallelogram b. rectangle c. triangle d. square

#### Solution:

The quadrilaterals parallelogram, square and a rectangle have two pairs of parallel sides each.

Angles in a square and a rectangle are right angles.

Parallelogram is the quadrilateral that has two pairs of parallel sides and does not have right angles.

2.
Which name does not suit the quadrilateral RSTU?

 a. Trapezoid b. Rectangle c. Rhombus d. None of the above

#### Solution:

The quadrilateral RSTU has two pairs of parallel sides.

So, it can be a rectangle or a rhombus.

A trapezoid has only one pair of parallel sides.

So, the quadrilateral RSTU cannot be a trapezoid.

3.
Which of the following statements is/are correct?
1. In a rhombus and a square, the diagonals bisect each other.
2. In a rhombus and a square, the diagonals are perpendicular.
3. All the sides in a rhombus and a square are equal.
4. Rhombus and square have equal measures of their diagonals.
 a. (1), (2) and (3) b. (2) and (3) c. Only (4) d. All the statements

#### Solution:

The diagonals bisect each other in both rhombus and square. So, statement 1 is correct.

Both in rhombus and square, the diagonals are perpendicular to each other. So, statement 2 is correct.

All sides of rhombus and square are equal. So, statement 3 is correct.

The measures of diagonals of a square are equal, but the measures of diagonals of a rhombus need not be equal. So, statement 4 is incorrect.

4.
Which of the following statements is/are incorrect?
1. A square is a rhombus but a rhombus is not a square.
2. Both square and rhombus have four congruent sides.
3. Diagonals bisect each other at right angles in both square and rhombus.
4. Measures of diagonals are equal in both square and rhombus.
 a. Only (4) b. (1) and (2) c. (2) and (3) d. (3) and (4)

#### Solution:

A square is a rhombus, but a rhombus need not be a square. So, statement 1 is correct.

Both square and rhombus have four congruent sides. So, statement 2 is correct.

In both rhombus and square, the diagonals bisect each other at right angles. So, statement 3 is correct.

The measures of diagonals in a square are equal, but not in a rhombus. So, statement 4 is incorrect.

5.
Which among the following statements is/are correct for a rhombus and a rectangle?
1. The diagonals bisect each other at right angles.
2. The opposite angles are equal.
3. The measures of diagonals are equal.
4. All sides are equal.
 a. 1 and 2 b. Only 2 c. 1 and 4 d. 3 and 4

#### Solution:

In a rhombus, diagonals bisect each other at right angles, but in a rectangle, diagonals do not intersect at right angles. So, statement 1 is incorrect.

Opposite angles are equal for both rhombus and rectangle. So, statement 2 is correct.

The measure of diagonals of a rectangle are equal, but the measures of diagonals of a rhombus need not be equal. So, statement 3 is not correct.

All sides of a rhombus are equal, but only opposite sides of a rectangle are equal. So, statement 4 is incorrect.

6.
Three angles of a quadrilateral are 109o, 71o, and 99o. What is the measure of the fourth angle?
 a. 279o b. 109o c. 69.75o d. 81o

#### Solution:

Sum of the four angle measures in a quadrilateral is 360o.

Let x be the measure of the fourth angle.

x + 109o + 71o + 99o = 360o

x + 279o = 360o

x = 360o - 279o
[Subtract 279 from both sides.]

x = 81o

So, the measure of the fourth angle is 81o.

7.
The measures of all the sides of the figure shown are equal and the adjacent sides are perpendicular to each other. Name the figure.

 a. Trapezoid b. Rectangle c. Square d. Rhombus

#### Solution:

The square is the only figure which has all four sides of equal measure, as well as adjacent sides that are perpendicular.

So, the quadrilateral is a square.

8.
ABCD is a square. What is the measure of $\angle$DOC?

 a. 60o b. 90o c. 45o d. 120o

#### Solution:

AC and BD are the diagonals of the square ABCD which intersect at the point O.

Diagonals of a square bisect each other at right angles.

So, DOC = 90o.

9.
ABCD is a rhombus. What is the measure of $\angle$DAB, if $\angle$ABC = 108o?

 a. 42o b. 144o c. 92o d. 72o

#### Solution:

In a rhombus, adjacent angles are supplementary.

ABC + DAB = 180o

DAB = 180o - ABC = 180o - 108o = 72o
[Substitute and simplify.]

10.
ABCD is a parallelogram of area 56 square units. The two diagonals AC and BD intersect each other at point O. Find the area of the shaded region.

 a. 42 square units b. 37 square units c. 47 square units d. 28 square units

#### Solution:

The two diagonals of a parallelogram bisect each other and divide the parallelogram into 4 triangular parts of equal area.

Area of the parallelogram = 56 square units

Area of ΔDOC = one fourth of area of ABCD = 56 / 4 = 14 square units

Area of the shaded region = the area of the parallelogram - the area of ΔDOC = 56 - 14 = 42 square units
[Substitute and simplify.]

The area of the shaded region is 42 square units.