Geometry Study Resources for Competitive Exams

Geometry

Q. In a coordinate plane, if the coordinates of two points are (2, 3) and (2, 7), what is the distance between these two points?

A. 4 units
B. 5 units
C. 3 units
D. 6 units

Solution

The distance between two points with the same x-coordinate is the difference in their y-coordinates: |7 - 3| = 4 units.

Correct Answer: A — 4 units

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Q. In a coordinate plane, if the coordinates of two points are (2, 3) and (2, 7), what is the slope of the line connecting them?

A. 0
B. Undefined
C. 1
D. -1

Solution

The slope is calculated as (y2 - y1) / (x2 - x1). Here, (7 - 3) / (2 - 2) is undefined because the denominator is 0.

Correct Answer: B — Undefined

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Q. In a coordinate plane, if the coordinates of two points on a line are (2, 3) and (4, 7), what is the slope of the line?

A. 2
B. 1
C. 3/2
D. 4/3

Solution

The slope is calculated as (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4/2 = 2.

Correct Answer: A — 2

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Q. In a coordinate plane, what is the distance between the points (1, 2) and (4, 6)?

A. 5 units
B. 4 units
C. 3 units
D. 6 units

Solution

Distance = √((4-1)² + (6-2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5 units.

Correct Answer: A — 5 units

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Q. In a coordinate plane, what is the distance between the points (3, 4) and (7, 1)?

A. 5 units
B. 4 units
C. 3 units
D. 6 units

Solution

The distance between two points (x1, y1) and (x2, y2) is given by the formula √((x2 - x1)² + (y2 - y1)²). Thus, distance = √((7 - 3)² + (1 - 4)²) = √(16 + 9) = √25 = 5 units.

Correct Answer: A — 5 units

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Q. In a coordinate plane, what is the equation of a circle with center at (3, -2) and radius 4?

A. (x - 3)² + (y + 2)² = 16
B. (x + 3)² + (y - 2)² = 16
C. (x - 3)² + (y - 2)² = 16
D. (x + 3)² + (y + 2)² = 16

Solution

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Correct Answer: A — (x - 3)² + (y + 2)² = 16

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Q. In a coordinate plane, what is the midpoint of the line segment connecting the points (1, 2) and (3, 4)?

A. (2, 3)
B. (1, 2)
C. (3, 4)
D. (4, 6)

Solution

The midpoint M is given by M = ((x1 + x2)/2, (y1 + y2)/2) = ((1 + 3)/2, (2 + 4)/2) = (2, 3).

Correct Answer: A — (2, 3)

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Q. In a coordinate plane, what is the slope of the line passing through the points (1, 2) and (3, 6)?

A. 2
B. 3
C. 1
D. 4

Solution

The slope m is calculated as m = (y2 - y1) / (x2 - x1) = (6 - 2) / (3 - 1) = 4 / 2 = 2.

Correct Answer: A — 2

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Q. In a coordinate plane, what is the slope of the line passing through the points (2, 3) and (4, 7)?

A. 2
B. 1
C. 0.5
D. 3

Solution

Slope = (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4 / 2 = 2.

Correct Answer: A — 2

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Q. In a coordinate system, if line 1 has the equation y = -3x + 4 and line 2 is parallel to it, what is the slope of line 2?

A. -3
B. 3
C. 1/3
D. 0

Solution

Parallel lines have the same slope. The slope of line 1 is -3, so the slope of line 2 is also -3.

Correct Answer: A — -3

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Q. In a coordinate system, if line A has the equation y = -1/2x + 3 and line B is parallel to line A, what is the slope of line B?

A. -1/2
B. 1/2
C. 2
D. -2

Solution

Parallel lines have the same slope, so the slope of line B is also -1/2.

Correct Answer: A — -1/2

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Q. In a coordinate system, if line A has the equation y = -1/2x + 4 and line B is parallel to line A, what is the slope of line B?

A. -1/2
B. 1/2
C. 2
D. 0

Solution

The slope of line B is the same as line A, which is -1/2.

Correct Answer: A — -1/2

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Q. In a coordinate system, if line A has the equation y = -1/2x + 4, what is the slope of a line parallel to line A?

A. 1/2
B. -1/2
C. 2
D. -2

Solution

Parallel lines have the same slope, so the slope of the parallel line is -1/2.

Correct Answer: B — -1/2

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Q. In a cyclic quadrilateral, if one angle is 80 degrees, what is the measure of the opposite angle?

A. 100 degrees
B. 80 degrees
C. 90 degrees
D. 70 degrees

Solution

In a cyclic quadrilateral, the opposite angles are supplementary. Therefore, if one angle is 80 degrees, the opposite angle is 180 - 80 = 100 degrees.

Correct Answer: A — 100 degrees

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Q. In a cyclic quadrilateral, what is the relationship between the opposite angles?

A. They are equal
B. They are supplementary
C. They are complementary
D. They are congruent

Solution

In a cyclic quadrilateral, the opposite angles are supplementary, meaning they add up to 180 degrees.

Correct Answer: B — They are supplementary

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Q. In a diagram where line AB is parallel to line CD and line EF is a transversal, if angle 1 is 70 degrees, what is the measure of angle 2, which is an alternate interior angle?

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

Solution

Since angle 1 and angle 2 are alternate interior angles formed by the transversal cutting through parallel lines, angle 2 is also 70 degrees.

Correct Answer: A — 70 degrees

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Q. In a diagram where two parallel lines are intersected by a transversal, if one of the corresponding angles measures 75 degrees, what is the measure of the other corresponding angle?

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

Solution

Corresponding angles are equal when two parallel lines are cut by a transversal. Therefore, the other corresponding angle also measures 75 degrees.

Correct Answer: A — 75 degrees

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Q. In a diagram with two parallel lines and a transversal, if angle 3 is 30 degrees, what is the measure of angle 4, which is an alternate interior angle?

A. 30 degrees
B. 150 degrees
C. 90 degrees
D. 60 degrees

Solution

Alternate interior angles are equal, so angle 4 = 30 degrees.

Correct Answer: A — 30 degrees

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Q. In a diagram with two parallel lines and a transversal, if angle 3 is 40 degrees, what is the measure of angle 4, which is an alternate interior angle?

A. 40 degrees
B. 140 degrees
C. 180 degrees
D. 90 degrees

Solution

Alternate interior angles are equal, so angle 4 also measures 40 degrees.

Correct Answer: A — 40 degrees

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Q. In a diagram with two parallel lines and a transversal, if one of the interior angles is 40 degrees, what is the measure of the corresponding angle?

A. 40 degrees
B. 140 degrees
C. 180 degrees
D. 90 degrees

Solution

Corresponding angles are equal when a transversal intersects two parallel lines. Therefore, the corresponding angle also measures 40 degrees.

Correct Answer: A — 40 degrees

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Q. In a diagram, if angle 1 and angle 2 are corresponding angles formed by a transversal intersecting two parallel lines, what can be concluded?

A. Angle 1 is greater than angle 2.
B. Angle 1 is less than angle 2.
C. Angle 1 is equal to angle 2.
D. Angle 1 and angle 2 are complementary.

Solution

Corresponding angles are equal when two parallel lines are cut by a transversal.

Correct Answer: C — Angle 1 is equal to angle 2.

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Q. In a figure with two parallel lines and a transversal, if one angle measures 30 degrees, what is the measure of the vertically opposite angle?

A. 30 degrees
B. 150 degrees
C. 90 degrees
D. 60 degrees

Solution

Vertically opposite angles are equal. Therefore, the vertically opposite angle also measures 30 degrees.

Correct Answer: A — 30 degrees

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Q. In a figure with two parallel lines cut by a transversal, if one angle measures 30 degrees, what is the measure of the vertically opposite angle?

A. 30 degrees
B. 150 degrees
C. 90 degrees
D. 60 degrees

Solution

Vertically opposite angles are equal, so the vertically opposite angle also measures 30 degrees.

Correct Answer: A — 30 degrees

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

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

Solution

Alternate interior angles are equal, so the other angle is also 75 degrees.

Correct Answer: A — 75 degrees

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Q. In a pair of parallel lines cut by a transversal, if one of the angles measures 110 degrees, what is the measure of the vertically opposite angle?

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

Solution

Vertically opposite angles are equal. Therefore, the vertically opposite angle also measures 110 degrees.

Correct Answer: B — 110 degrees

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Q. In a pair of parallel lines cut by a transversal, if one of the interior angles is 120°, what is the measure of the other interior angle on the same side of the transversal?

A. 60°
B. 120°
C. 180°
D. 90°

Solution

The two interior angles on the same side of the transversal are supplementary, so the other angle is 180° - 120° = 60°.

Correct Answer: A — 60°

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Q. In a pair of parallel lines cut by a transversal, if one of the interior angles is 120 degrees, what is the measure of the corresponding exterior angle?

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

Solution

The corresponding exterior angle is supplementary to the interior angle. Therefore, it measures 180 - 120 = 60 degrees.

Correct Answer: A — 60 degrees

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Q. In a pair of parallel lines cut by a transversal, if one of the interior angles is 120°, what is the measure of the corresponding angle?

A. 60°
B. 120°
C. 180°
D. 90°

Solution

Corresponding angles are equal, so the corresponding angle is also 120°.

Correct Answer: B — 120°

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Q. In a pair of parallel lines cut by a transversal, if one of the interior angles is 40 degrees, what is the measure of the adjacent interior angle?

A. 40 degrees
B. 140 degrees
C. 180 degrees
D. 90 degrees

Solution

Adjacent interior angles are supplementary, so the adjacent angle measures 140 degrees.

Correct Answer: B — 140 degrees

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Q. In a pair of parallel lines cut by a transversal, if one of the interior angles is 50 degrees, what is the measure of the corresponding angle?

A. 50 degrees
B. 130 degrees
C. 90 degrees
D. 180 degrees

Solution

Corresponding angles are equal when two parallel lines are cut by a transversal. Therefore, the corresponding angle also measures 50 degrees.

Correct Answer: A — 50 degrees

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Geometry

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