Geometry Study Resources for Competitive Exams

Geometry

Q. If two parallel lines are cut by a transversal and one of the interior angles is 40 degrees, what is the measure of the same-side interior angle?

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

Solution

Same-side interior angles are supplementary, so the angle measures 140 degrees.

Correct Answer: B — 140 degrees

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Q. If two parallel lines are cut by a transversal and one of the same-side exterior angles is 110 degrees, what is the measure of the other same-side exterior angle?

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

Solution

Same-side exterior angles are supplementary when two parallel lines are cut by a transversal. Therefore, the other angle measures 180 - 110 = 70 degrees.

Correct Answer: A — 70 degrees

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Q. If two parallel lines are cut by a transversal and one of the same-side interior angles is 130 degrees, what is the measure of the other same-side interior angle?

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

Solution

Same-side interior angles are supplementary, so the other angle measures 180 - 130 = 50 degrees.

Correct Answer: A — 50 degrees

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Q. If two parallel lines are cut by a transversal and one of the same-side interior angles is 40 degrees, what is the measure of the other same-side interior angle?

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

Solution

Same-side interior angles are supplementary when two parallel lines are cut by a transversal. Therefore, the other angle measures 180 - 40 = 140 degrees.

Correct Answer: B — 140 degrees

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Q. If two parallel lines are cut by a transversal and one of the same-side interior angles is 65 degrees, what is the measure of the other same-side interior angle?

A. 115 degrees
B. 65 degrees
C. 180 degrees
D. 90 degrees

Solution

Same-side interior angles are supplementary, so the other angle = 180 - 65 = 115 degrees.

Correct Answer: A — 115 degrees

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Q. If two parallel lines are cut by a transversal and 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, so the other angle measures 105 degrees.

Correct Answer: B — 105 degrees

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Q. If two parallel lines are cut by a transversal and one of the same-side interior angles is 120 degrees, what is the measure of the other same-side interior angle?

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

Solution

Same-side interior angles are supplementary, so 180 - 120 = 60 degrees.

Correct Answer: A — 60 degrees

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Q. If two parallel lines are cut by a transversal and the sum of the interior angles on the same side of the transversal is 180 degrees, what can be concluded?

A. The lines are not parallel.
B. The lines are perpendicular.
C. The angles are equal.
D. The angles are supplementary.

Solution

Interior angles on the same side of the transversal are supplementary when two parallel lines are cut by a transversal.

Correct Answer: D — The angles are supplementary.

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Q. If two parallel lines are intersected by a transversal and one of the exterior angles is 120 degrees, what is the measure of the corresponding interior angle?

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

Solution

The corresponding interior angle is equal to 180 - 120 = 60 degrees.

Correct Answer: A — 60 degrees

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Q. If two parallel lines are intersected by a transversal and one of the interior angles is 70 degrees, what is the measure of the other interior angle on the same side of the transversal?

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

Solution

Interior angles on the same side of the transversal are supplementary, so 180 - 70 = 110 degrees.

Correct Answer: B — 110 degrees

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Q. If two parallel lines are intersected by a transversal and one of the interior angles measures 70 degrees, what is the measure of the other interior angle on the same side of the transversal?

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

Solution

Interior angles on the same side of the transversal are supplementary, so 180 - 70 = 110 degrees.

Correct Answer: B — 110 degrees

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Q. If two parallel lines are intersected by a transversal, and one of the corresponding angles measures 45 degrees, what is the measure of the other corresponding angle?

A. 45 degrees
B. 135 degrees
C. 90 degrees
D. 180 degrees

Solution

Corresponding angles are equal when two parallel lines are cut by a transversal, so the other corresponding angle also measures 45 degrees.

Correct Answer: A — 45 degrees

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Q. If two parallel lines are intersected by a transversal, and one of the exterior angles is 120 degrees, what is the measure of the opposite exterior angle?

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

Solution

Opposite exterior angles are equal, so the measure is also 120 degrees.

Correct Answer: B — 120 degrees

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Q. If two parallel lines are represented by the equations y = 2x + 3 and y = 2x - 5, what is the distance between them?

A. 8/√5
B. 5/√5
C. 3/√5
D. 10/√5

Solution

The distance between two parallel lines y = mx + b1 and y = mx + b2 is |b2 - b1| / √(1 + m^2). Here, |(-5) - 3| / √(1 + 2^2) = 8/√5.

Correct Answer: A — 8/√5

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Q. If two parallel lines are represented by the equations y = 3x + 1 and y = 3x - 4, what is the distance between these two lines?

A. 5/√10
B. 5/√13
C. 5/√3
D. 5/√2

Solution

The distance d between two parallel lines of the form y = mx + b1 and y = mx + b2 is given by d = |b2 - b1| / √(1 + m^2). Here, d = |(-4) - 1| / √(1 + 3^2) = 5 / √10.

Correct Answer: B — 5/√13

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Q. If two parallel lines are represented by the equations y = 3x + 1 and y = 3x - 4, what is the distance between them?

A. 5
B. 4/√10
C. 3/√10
D. 7

Solution

The distance between two parallel lines y = mx + b1 and y = mx + b2 is |b2 - b1| / √(1 + m^2). Here, |(-4) - 1| / √(1 + 3^2) = 5 / √10 = 5/√10.

Correct Answer: B — 4/√10

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Q. If two parallel lines are represented by the equations y = 3x + 2 and y = 3x - 4, what is the distance between these two lines?

A. 6/√10
B. 2/√10
C. 4/√10
D. 8/√10

Solution

The distance between two parallel lines of the form y = mx + b1 and y = mx + b2 is given by |b2 - b1| / √(1 + m^2). Here, |(-4) - 2| / √(1 + 3^2) = 6 / √10.

Correct Answer: A — 6/√10

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Q. If two parallel lines are represented by the equations y = 4x + 1 and y = 4x - 3, what is the distance between them?

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

Solution

The distance between two parallel lines of the form y = mx + b1 and y = mx + b2 is given by |b2 - b1| / sqrt(1 + m^2). Here, |(-3) - 1| / sqrt(1 + 4) = 4 / sqrt(5) which approximates to 1.

Correct Answer: B — 2

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Q. If two sides of a triangle are 7 cm and 10 cm, what is the range of possible lengths for the third side?

A. 3 cm to 17 cm
B. 4 cm to 16 cm
C. 5 cm to 15 cm
D. 6 cm to 13 cm

Solution

By the triangle inequality theorem, the length of the third side must be greater than the difference of the other two sides and less than their sum: |7 - 10| < x < 7 + 10, which gives 3 < x < 17.

Correct Answer: A — 3 cm to 17 cm

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Q. If two similar triangles have a ratio of 2:3, what is the ratio of their areas?

A. 4:9
B. 2:3
C. 3:4
D. 1:2

Solution

The ratio of areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, (2:3)² = 4:9.

Correct Answer: A — 4:9

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Q. If two tangents are drawn from a point outside a circle to the circle, and the lengths of the tangents are equal, what can be said about the point and the circle?

A. The point is inside the circle
B. The point is outside the circle
C. The point is on the circle
D. The point is the center of the circle

Solution

The point is outside the circle, and the lengths of the tangents from a point outside a circle are equal.

Correct Answer: B — The point is outside the circle

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Q. If two tangents are drawn from a point outside a circle to the circle, what is the relationship between the lengths of the tangents?

A. They are equal
B. One is longer
C. They are perpendicular
D. They are complementary

Solution

The lengths of the tangents drawn from an external point to a circle are always equal.

Correct Answer: A — They are equal

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Q. If two tangents are drawn from a point outside a circle, what is the relationship between the lengths of the tangents?

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

Solution

The lengths of the tangents drawn from an external point to a circle are equal.

Correct Answer: A — They are equal

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Q. If two tangents are drawn from an external point to a circle, what can be said about the lengths of the tangents?

A. They are equal
B. They are unequal
C. One is longer
D. One is shorter

Solution

The lengths of the tangents drawn from an external point to a circle are equal.

Correct Answer: A — They are equal

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Q. If two triangles are congruent by the ASA criterion, what can be concluded about their sides?

A. They are all equal
B. They are all different
C. Some are equal
D. Cannot be determined

Solution

By the ASA (Angle-Side-Angle) criterion, if two triangles are congruent, then all their corresponding sides are equal.

Correct Answer: A — They are all equal

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Q. If two triangles are congruent by the ASA criterion, which of the following must be true?

A. Their corresponding sides are equal
B. Their corresponding angles are equal
C. Their areas are equal
D. All of the above

Solution

By the ASA criterion, if two triangles have two equal angles and the included side equal, then all corresponding sides and angles are equal.

Correct Answer: A — Their corresponding sides are equal

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Q. If two triangles are congruent by the SSS criterion, what can be said about their corresponding angles?

A. They are equal
B. They are supplementary
C. They are complementary
D. They are not related

Solution

If two triangles are congruent by SSS (Side-Side-Side), then their corresponding angles are equal.

Correct Answer: A — They are equal

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Q. If two triangles are congruent by the SSS criterion, which of the following must be true?

A. Their angles are equal
B. Their sides are equal
C. Their areas are equal
D. All of the above

Solution

By the SSS criterion, if two triangles have all three sides equal, then their corresponding sides are equal.

Correct Answer: B — Their sides are equal

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Q. If two triangles are congruent, what can be said about their corresponding angles?

A. They are equal
B. They are different
C. They are proportional
D. They are supplementary

Solution

Congruent triangles have corresponding angles that are equal.

Correct Answer: A — They are equal

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Q. If two triangles are congruent, what can be said about their corresponding sides?

A. They are equal.
B. They are proportional.
C. They are similar.
D. They are not related.

Solution

In congruent triangles, corresponding sides are equal.

Correct Answer: A — They are equal.

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Geometry

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