Geometry Study Resources for Competitive Exams

Geometry

Q. In a pair of parallel lines cut by a transversal, if one of the interior angles is 55 degrees, what is the measure of the corresponding exterior angle?

A. 125 degrees
B. 55 degrees
C. 180 degrees
D. 90 degrees

Solution

The corresponding exterior angle is supplementary to the interior angle. Thus, it measures 180 - 55 = 125 degrees.

Correct Answer: A — 125 degrees

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Q. In a pair of parallel lines cut by a transversal, if one of the same-side interior angles is 75 degrees, what is the measure of the other same-side interior angle?

A. 75 degrees
B. 105 degrees
C. 90 degrees
D. 180 degrees

Solution

Same-side interior angles are supplementary when two parallel lines are cut by a transversal. Therefore, the other angle measures 180 - 75 = 105 degrees.

Correct Answer: B — 105 degrees

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Q. In a parallelogram, if one angle measures 70 degrees, what are the measures of the other three angles?

A. 70, 110, 70 degrees
B. 70, 70, 110 degrees
C. 110, 70, 110 degrees
D. 90, 90, 90 degrees

Solution

In a parallelogram, opposite angles are equal and adjacent angles are supplementary. Therefore, if one angle is 70 degrees, the opposite angle is also 70 degrees, and the adjacent angles are 110 degrees each.

Correct Answer: B — 70, 70, 110 degrees

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Q. In a parallelogram, if one angle measures 70 degrees, what is the measure of the opposite angle?

A. 70 degrees
B. 110 degrees
C. 90 degrees
D. 180 degrees

Solution

In a parallelogram, opposite angles are equal. Therefore, if one angle is 70 degrees, the opposite angle is also 70 degrees.

Correct Answer: A — 70 degrees

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Q. In a quadrilateral ABCD, if angle A = 70 degrees, angle B = 110 degrees, and angle C = 90 degrees, what is the measure of angle D?

A. 80 degrees
B. 90 degrees
C. 100 degrees
D. 110 degrees

Solution

The sum of the angles in a quadrilateral is 360 degrees. Therefore, angle D = 360 - (70 + 110 + 90) = 360 - 270 = 90 degrees.

Correct Answer: A — 80 degrees

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Q. In a quadrilateral ABCD, if angle A = 70°, angle B = 110°, and angle C = 90°, what is the measure of angle D?

A. 80°
B. 90°
C. 100°
D. 110°

Solution

The sum of the angles in a quadrilateral is 360°. Therefore, angle D = 360° - (70° + 110° + 90°) = 360° - 270° = 90°.

Correct Answer: A — 80°

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Q. In a rectangle, if the length is 10 cm and the width is 5 cm, what is the area?

A. 15 cm²
B. 50 cm²
C. 30 cm²
D. 100 cm²

Solution

The area of a rectangle is calculated by multiplying the length by the width. Area = length * width = 10 cm * 5 cm = 50 cm².

Correct Answer: B — 50 cm²

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Q. In a rectangle, if the length is 10 cm and the width is 5 cm, what is the perimeter?

A. 30 cm
B. 25 cm
C. 20 cm
D. 15 cm

Solution

The perimeter of a rectangle is given by the formula: P = 2(length + width). Here, P = 2(10 + 5) = 30 cm.

Correct Answer: A — 30 cm

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Q. In a rectangle, if the length is 8 cm and the width is 3 cm, what is the perimeter?

A. 22 cm
B. 24 cm
C. 20 cm
D. 26 cm

Solution

The perimeter of a rectangle is given by the formula: P = 2 * (length + width). Here, P = 2 * (8 + 3) = 22 cm.

Correct Answer: A — 22 cm

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Q. In a rectangle, if the length is 8 cm and the width is 5 cm, what is the area?

A. 40 cm²
B. 30 cm²
C. 50 cm²
D. 60 cm²

Solution

The area of a rectangle is calculated using the formula length × width. Thus, the area = 8 cm × 5 cm = 40 cm².

Correct Answer: A — 40 cm²

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Q. In a rectangle, if the length is 8 cm and the width is 5 cm, what is the perimeter?

A. 26 cm
B. 40 cm
C. 30 cm
D. 20 cm

Solution

The perimeter of a rectangle is given by the formula: P = 2(l + w). Here, l = 8 cm and w = 5 cm. P = 2(8 + 5) = 26 cm.

Correct Answer: A — 26 cm

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Q. In a rectangle, if the length is doubled and the width remains the same, how does the area change?

A. It remains the same
B. It doubles
C. It triples
D. It quadruples

Solution

If the length is doubled, the new area becomes 2 × length × width, which is double the original area.

Correct Answer: B — It doubles

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Q. In a rhombus with diagonals of lengths 10 and 24, what is the area?

A. 120
B. 140
C. 160
D. 180

Solution

The area of a rhombus can be calculated using the formula: Area = 1/2 * d1 * d2. Therefore, area = 1/2 * 10 * 24 = 120.

Correct Answer: A — 120

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Q. In a rhombus, if one angle measures 120 degrees, what are the measures of the other three angles?

A. 120, 60, 120
B. 60, 120, 60
C. 120, 120, 60
D. 60, 60, 120

Solution

In a rhombus, opposite angles are equal and adjacent angles are supplementary. Thus, if one angle is 120 degrees, the opposite angle is also 120 degrees, and the adjacent angles are 60 degrees.

Correct Answer: A — 120, 60, 120

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Q. In a rhombus, if one angle measures 60 degrees, what is the measure of the adjacent angle?

A. 120 degrees
B. 60 degrees
C. 90 degrees
D. 180 degrees

Solution

In a rhombus, adjacent angles are supplementary. Therefore, if one angle is 60 degrees, the adjacent angle is 180 - 60 = 120 degrees.

Correct Answer: A — 120 degrees

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Q. In a rhombus, if one diagonal measures 10 cm and the other measures 24 cm, what is the area?

A. 120 cm²
B. 240 cm²
C. 60 cm²
D. 100 cm²

Solution

The area of a rhombus can be calculated using the formula (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals. Here, area = (10 cm * 24 cm) / 2 = 120 cm².

Correct Answer: A — 120 cm²

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Q. In a rhombus, if one diagonal measures 10 cm and the other measures 24 cm, what is the area of the rhombus?

A. 120 cm²
B. 240 cm²
C. 60 cm²
D. 100 cm²

Solution

The area of a rhombus can be calculated using the formula Area = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals. Here, Area = (10 cm × 24 cm) / 2 = 120 cm².

Correct Answer: A — 120 cm²

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Q. In a rhombus, if one diagonal measures 12 cm and the other measures 16 cm, what is the area of the rhombus?

A. 96 cm²
B. 48 cm²
C. 192 cm²
D. 64 cm²

Solution

The area of a rhombus can be calculated using the formula: Area = (d1 × d2) / 2. Here, Area = (12 cm × 16 cm) / 2 = 96 cm².

Correct Answer: A — 96 cm²

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Q. In a right triangle inscribed in a circle, what is the relationship between the hypotenuse and the diameter of the circle?

A. They are equal
B. The hypotenuse is longer
C. The hypotenuse is shorter
D. They are unrelated

Solution

In a right triangle inscribed in a circle, the hypotenuse is equal to the diameter of the circle.

Correct Answer: A — They are equal

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Q. In a right triangle, if one angle is 30 degrees and the hypotenuse is 10 units, what is the length of the side opposite the 30-degree angle?

A. 5 units
B. 10 units
C. √3 units
D. √2 units

Solution

In a 30-60-90 triangle, the side opposite the 30-degree angle is half the hypotenuse. Thus, it is 10/2 = 5 units.

Correct Answer: A — 5 units

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Q. In a right triangle, if one angle is 30 degrees, what is the measure of the angle opposite the side that is half the length of the hypotenuse?

A. 30 degrees
B. 60 degrees
C. 90 degrees
D. 45 degrees

Solution

In a 30-60-90 triangle, the side opposite the 30-degree angle is half the hypotenuse.

Correct Answer: A — 30 degrees

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Q. In a right triangle, if one angle is 30 degrees, what is the measure of the angle opposite the side that is half the hypotenuse?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees

Solution

In a 30-60-90 triangle, the side opposite the 30-degree angle is half the hypotenuse.

Correct Answer: A — 30 degrees

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Q. In a right triangle, if one angle is 30 degrees, what is the ratio of the lengths of the sides opposite to the 30 degrees and 90 degrees angles?

A. 1:2
B. 1:√3
C. 1:1
D. 2:1

Solution

In a 30-60-90 triangle, the ratio of the sides opposite the 30° and 90° angles is 1:2.

Correct Answer: A — 1:2

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Q. In a right triangle, if one angle is 30°, what is the measure of the other non-right angle?

A. 30°
B. 45°
C. 60°
D. 90°

Solution

In a right triangle, the sum of the angles is 180°. Therefore, the other angle = 180° - 90° - 30° = 60°.

Correct Answer: C — 60°

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Q. In a right triangle, if one angle is 30°, what is the ratio of the lengths of the sides opposite to the 30° and 90° angles?

A. 1:2
B. 1:√3
C. 1:1
D. 2:1

Solution

In a 30-60-90 triangle, the side opposite the 30° angle is half the hypotenuse, giving a ratio of 1:2.

Correct Answer: A — 1:2

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Q. In a right triangle, if one angle is 45 degrees and the hypotenuse is 10 cm, what is the length of each leg?

A. 5√2 cm
B. 10 cm
C. 5 cm
D. 7.5 cm

Solution

In a 45-45-90 triangle, the legs are equal and each leg = hypotenuse/√2 = 10/√2 = 5√2 cm.

Correct Answer: A — 5√2 cm

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Q. In a right triangle, if one angle is 45 degrees, what are the measures of the other two angles?

A. 45, 90
B. 30, 60
C. 60, 30
D. 90, 45

Solution

In a right triangle, the sum of the angles is 180 degrees. Therefore, the other angle must also be 45 degrees.

Correct Answer: A — 45, 90

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Q. In a right triangle, if one angle is 45 degrees, what is the ratio of the lengths of the legs?

A. 1:2
B. 1:1
C. 2:1
D. √2:1

Solution

In a 45-45-90 triangle, the legs are equal, so the ratio is 1:1.

Correct Answer: B — 1:1

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Q. In a right triangle, if one angle is 45 degrees, what is the ratio of the lengths of the sides opposite and adjacent to this angle?

A. 1:1
B. 1:√2
C. √2:1
D. 2:1

Solution

In a 45-45-90 triangle, the sides opposite the 45-degree angles are equal, hence the ratio is 1:1.

Correct Answer: A — 1:1

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Q. In a right triangle, if one angle is 45°, what is the ratio of the lengths of the sides opposite and adjacent to this angle?

A. 1:1
B. 1:√2
C. √2:1
D. 2:1

Solution

In a 45°-45°-90° triangle, the sides opposite and adjacent are equal, so the ratio is 1:1.

Correct Answer: A — 1:1

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Geometry

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