Geometry Study Resources for Competitive Exams

Geometry

Q. What is the length of the side of a square with vertices at (1, 1), (1, 5), (5, 1), and (5, 5)?

A. 4
B. 5
C. 6
D. 7

Solution

Length of side = 5 - 1 = 4.

Correct Answer: A — 4

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Q. What is the length of the tangent from a point P outside a circle to the point of tangency T if the radius of the circle is 5 cm and the distance from P to the center O of the circle is 13 cm?

A. 12 cm
B. 10 cm
C. 8 cm
D. 6 cm

Solution

Using the tangent-secant theorem, PT² = PO² - OT². Thus, PT² = 13² - 5² = 169 - 25 = 144, so PT = 12 cm.

Correct Answer: A — 12 cm

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Q. What is the measure of an angle formed by a tangent and a chord drawn from the point of contact?

A. It is equal to the angle subtended by the chord at the center.
B. It is equal to half the angle subtended by the chord at the circumference.
C. It is equal to the angle subtended by the tangent at the center.
D. It is always 90 degrees.

Solution

The angle formed by a tangent and a chord is equal to half the angle subtended by the chord at the circumference.

Correct Answer: B — It is equal to half the angle subtended by the chord at the circumference.

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Q. What is the measure of an angle formed by a tangent and a chord through the point of contact?

A. Equal to the angle subtended by the chord at the center.
B. Equal to the angle subtended by the chord at the circumference.
C. Supplementary to the angle subtended by the chord.
D. Equal to twice the angle subtended by the chord.

Solution

The angle formed by a tangent and a chord through the point of contact is equal to the angle subtended by the chord at the circumference.

Correct Answer: B — Equal to the angle subtended by the chord at the circumference.

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Q. What is the measure of an angle formed by two tangents drawn from a point outside a circle?

A. Half the difference of the intercepted arcs.
B. Half the sum of the intercepted arcs.
C. Equal to the intercepted arc.
D. Equal to the radius.

Solution

The angle formed by two tangents from a point outside a circle is half the difference of the measures of the intercepted arcs.

Correct Answer: A — Half the difference of the intercepted arcs.

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Q. What is the measure of an exterior angle of a triangle if the two remote interior angles measure 40 degrees and 60 degrees?

A. 80 degrees
B. 100 degrees
C. 120 degrees
D. 140 degrees

Solution

The exterior angle is equal to the sum of the two remote interior angles. Therefore, the exterior angle measures 40 + 60 = 100 degrees.

Correct Answer: B — 100 degrees

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Q. What is the measure of an exterior angle of a triangle if the two remote interior angles are 45 degrees and 55 degrees?

A. 90 degrees
B. 100 degrees
C. 110 degrees
D. 120 degrees

Solution

The exterior angle is equal to the sum of the two remote interior angles. Therefore, it is 45 + 55 = 100 degrees.

Correct Answer: B — 100 degrees

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Q. What is the measure of an exterior angle of a triangle if the two remote interior angles measure 50 degrees and 60 degrees?

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

Solution

The exterior angle is equal to the sum of the two remote interior angles. Therefore, it is 50 + 60 = 110 degrees.

Correct Answer: B — 80 degrees

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Q. What is the measure of an inscribed angle that intercepts an arc of 80 degrees?

A. 40 degrees
B. 80 degrees
C. 160 degrees
D. 20 degrees

Solution

The measure of an inscribed angle is half the measure of the intercepted arc. Therefore, it is 80 degrees / 2 = 40 degrees.

Correct Answer: A — 40 degrees

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Q. What is the measure of angle 3 if angle 1 is 45 degrees and angle 3 is an alternate exterior angle to angle 1?

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

Solution

Alternate exterior angles are equal, so angle 3 is also 45 degrees.

Correct Answer: A — 45 degrees

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Q. What is the measure of angle A if lines AB and CD are parallel and angle B is 70 degrees?

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

Solution

Angle A is equal to angle B because they are corresponding angles, so angle A = 70 degrees.

Correct Answer: B — 110 degrees

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Q. What is the measure of angle A if lines AB and CD are parallel and angle B is 70 degrees, where angle A is the corresponding angle to angle B?

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

Solution

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

Correct Answer: A — 70 degrees

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Q. What is the measure of angle x if two parallel lines are cut by a transversal and angle x is an exterior angle that is 40 degrees?

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

Solution

Exterior angles formed by a transversal with parallel lines are supplementary to the interior angles. Therefore, angle x = 180 - 40 = 140 degrees.

Correct Answer: B — 140 degrees

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Q. What is the measure of angle x if two parallel lines are cut by a transversal and one of the corresponding angles is 75 degrees?

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

Solution

Corresponding angles are equal, so angle x is also 75 degrees.

Correct Answer: A — 75 degrees

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Q. What is the measure of angle x if two parallel lines are cut by a transversal and angle x is an exterior angle that is supplementary to an interior angle measuring 120 degrees?

A. 60 degrees
B. 120 degrees
C. 90 degrees
D. 30 degrees

Solution

Exterior angles are supplementary to interior angles. Therefore, x = 180 - 120 = 60 degrees.

Correct Answer: A — 60 degrees

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Q. What is the measure of each angle formed by two parallel lines cut by a transversal if one angle measures 75 degrees?

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

Solution

The corresponding angle will measure 75 degrees, and the alternate interior angle will measure 105 degrees.

Correct Answer: B — 105 degrees

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Q. What is the measure of each angle in a pair of corresponding angles when two parallel lines are cut by a transversal?

A. They are always equal.
B. They are always supplementary.
C. They are always complementary.
D. They can be any value.

Solution

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

Correct Answer: A — They are always equal.

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Q. What is the measure of each angle in an equilateral triangle?

A. 60°
B. 45°
C. 90°
D. 30°

Solution

In an equilateral triangle, each angle measures 60°.

Correct Answer: A — 60°

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Q. What is the measure of each interior angle of a regular hexagon?

A. 120 degrees
B. 90 degrees
C. 60 degrees
D. 150 degrees

Solution

The measure of each interior angle of a regular hexagon is given by (n-2) × 180/n, where n is the number of sides. For a hexagon, n = 6, so each angle is (6-2) × 180/6 = 120 degrees.

Correct Answer: A — 120 degrees

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Q. What is the measure of each interior angle of a regular pentagon?

A. 108 degrees
B. 120 degrees
C. 90 degrees
D. 135 degrees

Solution

The measure of each interior angle of a regular polygon can be calculated using the formula [(n-2) × 180] / n. For a pentagon (n=5), each angle = [(5-2) × 180] / 5 = 108 degrees.

Correct Answer: A — 108 degrees

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Q. What is the measure of the angle formed by a transversal that intersects two parallel lines, if one of the interior angles measures 50 degrees?

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

Solution

Interior angles on the same side of the transversal are supplementary. Therefore, the angle formed is 180 - 50 = 130 degrees.

Correct Answer: B — 130 degrees

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Q. What is the measure of the angle subtended by a diameter at any point on the circle?

A. 90 degrees
B. 60 degrees
C. 45 degrees
D. 180 degrees

Solution

According to the inscribed angle theorem, the angle subtended by a diameter at any point on the circle is always 90 degrees.

Correct Answer: A — 90 degrees

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Q. What is the measure of the angle subtended by a diameter of a circle at any point on the circle?

A. 90 degrees
B. 60 degrees
C. 45 degrees
D. 180 degrees

Solution

According to the inscribed angle theorem, the angle subtended by a diameter at any point on the circle is always 90 degrees.

Correct Answer: A — 90 degrees

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Q. What is the measure of the angle subtended by an arc at the center of a circle compared to the angle subtended at any point on the remaining part of the circle?

A. Half the angle at the center
B. Equal to the angle at the center
C. Twice the angle at the center
D. None of the above

Solution

The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the remaining part of the circle.

Correct Answer: A — Half the angle at the center

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Q. What is the measure of the angle subtended by an arc of a circle at the center if the arc length is 10 units and the radius is 5 units?

A. 30 degrees
B. 60 degrees
C. 90 degrees
D. 120 degrees

Solution

Arc length = rθ, where θ is in radians. Thus, 10 = 5θ, so θ = 2 radians. Converting to degrees: θ = 2 * (180/π) ≈ 114.6 degrees.

Correct Answer: C — 90 degrees

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Q. What is the measure of the central angle that subtends an arc of 120 degrees in a circle?

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

Solution

The measure of the central angle is equal to the measure of the arc it subtends, which is 120 degrees.

Correct Answer: B — 120 degrees

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Q. What is the measure of the central angle that subtends an arc of length 5 cm in a circle of radius 10 cm?

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

Solution

The formula for the arc length is L = rθ, where θ is in radians. Thus, θ = L/r = 5/10 = 0.5 radians. Converting to degrees, θ = 0.5 * (180/π) ≈ 28.65 degrees, which is closest to 30 degrees.

Correct Answer: B — 60 degrees

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Q. What is the measure of the corresponding angle if one angle measures 75 degrees and the lines are parallel?

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.

Correct Answer: A — 75 degrees

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Q. What is the measure of the corresponding angle if one of the angles is 75 degrees and the lines are parallel?

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

Solution

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

Correct Answer: A — 75 degrees

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Q. What is the measure of the corresponding angle if one of the parallel lines is cut by a transversal and one angle measures 45 degrees?

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

Solution

Corresponding angles are equal, so the measure is also 45 degrees.

Correct Answer: A — 45 degrees

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Geometry

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