Geometry Practice Test: Triangle Congruence and Similarity
- Triangle Congruence and Similarity
- 15 questions
- 18 minutes
These questions cover congruence criteria, CPCTC, isosceles triangles, similarity criteria and proportional sides. Always match vertices in the order they are written, such as △ABC ≅ △DEF, before comparing sides.
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Question 1
Two triangles have three pairs of corresponding parts congruent.
Which combination does NOT guarantee that the triangles are congruent?
Correct answer: D
Two sides and a non-included angle (SSA) can fit two different triangles, so it is not a congruence criterion. SSS, SAS and AAS all guarantee congruence.
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Question 2
△ABC ≅ △DEF.
Which side must be congruent to side BC?
Correct answer: B
The order of the letters matches the corresponding vertices: B ↔ E and C ↔ F. So side BC corresponds to side EF.
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Question 3
△PQR ≅ △XYZ, m∠Q = 48° and m∠R = 67°.
What is m∠X?
Correct answer: B
∠X corresponds to ∠P, and m∠P = 180° − 48° − 67° = 65°. Congruent triangles have congruent corresponding angles, so m∠X = 65°.
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Question 4
In quadrilateral ABCD, side AB is parallel to and congruent to side CD. Diagonal AC is drawn.
Which criterion proves that △ABC ≅ △CDA?
Correct answer: C
AB ≅ CD is given, ∠BAC ≅ ∠DCA because they are alternate interior angles for the parallel sides, and AC ≅ CA by the Reflexive Property. The angle lies between the two sides, so this is SAS; SSS would need BC ≅ DA, which is not known yet.
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Question 5
A proof has just shown that △ABC ≅ △DEF by ASA. The next statement is ∠B ≅ ∠E.
Which reason justifies this statement?
Correct answer: A
Once two triangles are proved congruent, every pair of corresponding parts is congruent, and this is stated as CPCTC. ∠B and ∠E are corresponding angles, so CPCTC is the reason.
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Question 6
In △ABC, AB = AC and m∠A = 40°.
What is m∠B?
Correct answer: B
The base angles opposite the congruent sides are congruent, so ∠B and ∠C share 180° − 40° = 140° equally: m∠B = 70°.
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Question 7
△ABC ~ △DEF, with AB = 6, BC = 8 and DE = 15.
What is EF?
Correct answer: C
The scale factor from △ABC to △DEF is 15 ÷ 6 = 2.5, so EF = 8 · 2.5 = 20. The value 17 comes from adding 9 to each side, but similar figures are related by multiplication, not addition.
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Question 8
In △ABC, segment DE is parallel to side BC, with AD = 4, DB = 6 and AE = 5.
What is the value of x = EC?
Correct answer: D
A segment parallel to one side divides the other two sides proportionally: 4/6 = 5/x, so 4x = 30 and x = 7.5. The value 10/3 comes from flipping the proportion (4/6 = x/5).
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Question 9
Two angles of triangle 1 measure 50° and 60°. Two angles of triangle 2 measure 60° and 70°.
Which statement is true?
Correct answer: A
The third angle of triangle 1 is 70° and the third angle of triangle 2 is 50°, so both triangles have angles 50°, 60° and 70°. Two pairs of congruent angles are enough for AA similarity; without side lengths, congruence cannot be shown.
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Question 10
Two similar triangles have corresponding sides in the ratio 2 : 5. The smaller triangle has an area of 12 square units.
What is the area of the larger triangle?
Correct answer: C
Areas of similar figures scale by the square of the side ratio: (5/2)² = 25/4, and 12 · 25/4 = 75. Answering 30 multiplies by the side ratio 5/2 only.
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Question 11
At the same time of day, a 6-foot person casts a 4-foot shadow and a tree casts a 30-foot shadow.
How tall is the tree?
Correct answer: D
The sun makes similar right triangles, so height/shadow is the same: 6/4 = h/30, which gives h = 45 feet. The value 20 feet comes from the inverted ratio 4/6.
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Question 12
A triangle has side lengths 4, 6 and 8.
Which triangle is similar to it?
Correct answer: A
Each side of 6, 9, 12 is 1.5 times the matching side of 4, 6, 8, so the triangles are similar by SSS similarity. Adding the same number to each side (as in 5, 7, 9) does not keep the ratios equal.
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Question 13
Which transformation does NOT always produce an image that is congruent to the original figure?
Correct answer: D
Reflections, rotations and translations are rigid motions, so they preserve all lengths and angles. A dilation with scale factor 3 triples every length, so the image is similar but not congruent.
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Question 14
△ABC and △DEF are right triangles with right angles at C and F. Hypotenuse AB ≅ hypotenuse DE, and leg AC ≅ leg DF.
Which criterion proves the triangles congruent?
Correct answer: B
A congruent hypotenuse and a congruent leg in two right triangles give congruence by Hypotenuse-Leg (HL). SAS does not fit, because the right angle is not between the hypotenuse and leg AC.
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Question 15
In △ABC, D and E are the midpoints of AB and AC. DE = 2x + 3 and BC = 5x − 1.
What is the length of DE?
Correct answer: C
A midsegment is half as long as the third side: 2(2x + 3) = 5x − 1, so x = 7. Then DE = 2(7) + 3 = 17 (and BC = 34); 7 is only the value of x.
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