Geometry (Grades 9–10) Midterm — Full-Course Review

Math Grades 9–10 30 multiple choice + 2 written

Free printable Geometry midterm (Grades 9–10): 30 multiple-choice questions plus 2 written responses covering Foundations: Points, Lines & Angles, Reasoning & Proof, Parallel Lines & Triangles, Congruence & Similarity, Right Triangles & Trigonometry, Circles, Area & Volume. Answer key included — customize it in the Midterm Maker.

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Directions: Read each question carefully and choose the best answer. Answer the written response in complete sentences.

Covers: Foundations: Points, Lines & Angles · Reasoning & Proof · Parallel Lines & Triangles · Congruence & Similarity · Right Triangles & Trigonometry · Circles, Area & Volume

Part 1: Multiple Choice

M(3, 5) is the midpoint of segment AB, and A is at (1, 2). What are the coordinates of B?
1. A. (5, 8)
2. B. (2, 3.5)
3. C. (-1, -1)
4. D. (4, 7)
Ray BD bisects angle ABC. The measure of angle ABD is (2x + 10)° and the measure of angle DBC is (4x - 20)°. What is the measure of angle ABC?
1. A. 40°
2. B. 80°
3. C. 160°
4. D. 15°
M is the midpoint of segment AB. AM = 3x + 4 and MB = 5x - 6. What is AB?
1. A. 76
2. B. 19
3. C. 38
4. D. 5
The supplement of an angle is three times its complement. What is the measure of the angle?
1. A. 45°
2. B. 67.5°
3. C. 60°
4. D. 22.5°
Two lines intersect. A pair of vertical angles measure (5x - 10)° and (3x + 30)°. What is the measure of each of these angles?
1. A. 90°
2. B. 20°
3. C. 100°
4. D. 45°
Which number is a counterexample to "If a number is prime, then it is odd"?
1. A. 9
2. B. 2
3. C. 3
4. D. 15
In a proof, the step "2(x + 3) = 14" is followed by "2x + 6 = 14." Which reason justifies the second step?
1. A. Reflexive Property of Equality
2. B. Distributive Property
3. C. Division Property of Equality
4. D. Subtraction Property of Equality
What is the converse of "If an animal is a dog, then it has four legs"?
1. A. An animal is a dog if and only if it has four legs.
2. B. If an animal is not a dog, then it does not have four legs.
3. C. If an animal has four legs, then it is a dog.
4. D. If an animal does not have four legs, then it is not a dog.
Given: "If it rains, then the game is canceled" and "It is raining." Which conclusion follows by the Law of Detachment?
1. A. It will stop raining.
2. B. The game is canceled.
3. C. The game is not canceled.
4. D. If the game is canceled, then it is raining.
Use inductive reasoning to find the next number: 3, 7, 11, 15, ...
1. A. 18
2. B. 30
3. C. 21
4. D. 19
Two sides of a triangle measure 7 and 12. Which inequality describes all possible lengths x of the third side?
1. A. 5 < x < 19
2. B. 7 < x < 12
3. C. 0 < x < 19
4. D. 5 ≤ x ≤ 19
Two parallel lines are cut by a transversal. One angle measures 65°. What is the measure of its corresponding angle?
1. A. 115°
2. B. 130°
3. C. 25°
4. D. 65°
The two remote interior angles of a triangle measure 40° and 75°. What is the measure of the exterior angle at the third vertex?
1. A. 115°
2. B. 65°
3. C. 245°
4. D. 35°
What is the measure of each exterior angle of a regular octagon?
1. A. 135°
2. B. 40°
3. C. 60°
4. D. 45°
Two parallel lines are cut by a transversal. A pair of alternate interior angles measure (3x + 15)° and (5x - 25)°. What is x?
1. A. 65
2. B. 20
3. C. 23.75
4. D. 5
Two similar rectangles have perimeters in the ratio 2:5. What is the ratio of their areas?
1. A. 4:10
2. B. 8:125
3. C. 2:5
4. D. 4:25
A 6-foot person casts a 4-foot shadow. At the same time, a tree casts a 22-foot shadow. How tall is the tree?
1. A. 132 feet
2. B. 24 feet
3. C. 33 feet
4. D. 14.7 feet
Which transformation produces a figure that is similar but NOT congruent to the original?
1. A. A dilation with scale factor 3
2. B. A reflection over the y-axis
3. C. A translation 5 units up
4. D. A rotation of 90°
Which of these is NOT a valid way to prove two triangles congruent?
1. A. SAS
2. B. SSS
3. C. ASA
4. D. SSA
In triangles PQR and XYZ, ∠P ≅ ∠X, side PQ ≅ side XY, and ∠Q ≅ ∠Y. Which postulate proves the triangles congruent?
1. A. SSS
2. B. HL
3. C. SAS
4. D. ASA
A right triangle has legs of 9 and 12. How long is the hypotenuse?
1. A. 15
2. B. √63
3. C. 21
4. D. 225
A right triangle has legs of 5 and 12. To the nearest tenth of a degree, what is the measure of the angle opposite the side of length 5?
1. A. 24.6°
2. B. 65.4°
3. C. 67.4°
4. D. 22.6°
A right triangle has legs of 7 and 24 and a hypotenuse of 25. Angle θ is opposite the side of length 7. What is cos θ?
1. A. 7/25
2. B. 25/24
3. C. 7/24
4. D. 24/25
From a point on level ground 40 feet from the base of a flagpole, the angle of elevation to the top is 32°. To the nearest tenth of a foot, how tall is the flagpole?
1. A. 25.0 feet
2. B. 64.0 feet
3. C. 21.2 feet
4. D. 33.9 feet
Each leg of a 45°-45°-90° triangle is 7 units long. How long is the hypotenuse?
1. A. 7√2
2. B. 49
3. C. 14
4. D. 7√3
What is the surface area of a sphere with radius 5?
1. A. 500π/3
2. B. 25π
3. C. 50π
4. D. 100π
A trapezoid has bases of 8 cm and 14 cm and a height of 5 cm. What is its area?
1. A. 55 square cm
2. B. 40 square cm
3. C. 110 square cm
4. D. 70 square cm
A pyramid has a square base with sides of 6 m and a height of 10 m. What is its volume?
1. A. 360 cubic meters
2. B. 240 cubic meters
3. C. 120 cubic meters
4. D. 60 cubic meters
An inscribed angle in a circle intercepts an arc of 110°. What is the measure of the inscribed angle?
1. A. 220°
2. B. 110°
3. C. 70°
4. D. 55°
A line is tangent to a circle. What is the measure of the angle between the tangent and the radius drawn to the point of tangency?
1. A. 180°
2. B. 60°
3. C. 90°
4. D. 45°

Part 2: Written Response

Lines m and n are parallel and are cut by transversal t. ∠1 measures (4x + 12)° and ∠2 measures (6x - 28)°, and they are alternate interior angles. ∠3 forms a linear pair with ∠2. Find x, m∠1, m∠2, and m∠3, and name the theorem you used. Show your work.
A grain silo is a cylinder with radius 10 ft and height 40 ft, topped by a half sphere (hemisphere) with radius 10 ft. Find the total volume in terms of π and to the nearest cubic foot. Show your work.

Answer Key — Geometry (Grades 9–10) Midterm — Full-Course Review

A — (5, 8) B = (2·3 - 1, 2·5 - 2) = (5, 8).
B — 80° 2x + 10 = 4x - 20 gives x = 15; each half is 40°, so 80°.
C — 38 3x + 4 = 5x - 6 gives x = 5; AM = 19, so AB = 38.
A — 45° 180 - a = 3(90 - a) gives 2a = 90, so a = 45.
A — 90° Vertical angles are equal: 5x - 10 = 3x + 30 gives x = 20, so 90°.
B — 2 2 is prime but even, so the statement is false.
B — Distributive Property Multiplying 2 by each term inside the parentheses uses the Distributive Property.
C — If an animal has four legs, then it is a dog. The converse switches the hypothesis and the conclusion.
B — The game is canceled. When the hypothesis is true, the conclusion must be true.
D — 19 The pattern adds 4 each time: 15 + 4 = 19.
A — 5 < x < 19 The third side must be between 12 - 7 and 12 + 7, exclusive.
D — 65° Corresponding angles formed by parallel lines are congruent.
A — 115° An exterior angle equals the sum of the two remote interior angles.
D — 45° Exterior angles sum to 360°: 360 ÷ 8 = 45.
B — 20 Alternate interior angles are congruent: 3x + 15 = 5x - 25, so x = 20.
D — 4:25 Areas compare as the square of the scale factor: (2/5)² = 4/25.
C — 33 feet Similar triangles: 6/4 = h/22, so h = 33.
A — A dilation with scale factor 3 A dilation by 3 keeps shape but triples lengths.
D — SSA Two sides and a non-included angle can form two different triangles.
D — ASA Two angles and the included side between them match.
A — 15 √(81 + 144) = √225 = 15.
D — 22.6° tan⁻¹(5/12) ≈ 22.6°.
D — 24/25 cosine = adjacent/hypotenuse = 24/25.
A — 25.0 feet height = 40 × tan 32° ≈ 25.0 feet.
A — 7√2 In a 45°-45°-90° triangle, hypotenuse = leg × √2.
D — 100π SA = 4πr² = 4π(25) = 100π.
A — 55 square cm A = (1/2)(8 + 14)(5) = 55.
C — 120 cubic meters V = (1/3)Bh = (1/3)(36)(10) = 120.
D — 55° An inscribed angle is half its intercepted arc.
C — 90° A tangent is perpendicular to the radius at the point of tangency.

Written Response — Look-Fors

Alternate Interior Angles Theorem: 4x + 12 = 6x - 28, so 2x = 40; substitute into both expressions; ∠3 is supplementary to ∠2. Final answers: x = 20, m∠1 = m∠2 = 92°, m∠3 = 88°.
Cylinder πr²h = π(100)(40) = 4000π; hemisphere (1/2)(4/3)πr³ = 2000π/3; add them to get 14000π/3. Final answer: 14000π/3, about 14,661 cubic feet.

Geometry (Grades 9–10) Midterm — Full-Course Review

Part 1: Multiple Choice

Part 2: Written Response

Written Response — Look-Fors

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