Congruent figures
Given: ΔABC ≅ ΔDEF
AB = 5, BC = 7, AC = 9
DE = 5, EF = 7
What theorem proves ΔABC ≅ ΔDEF?
A) SSS
B) SAS
C) ASA
D) AAS
The correct answer is B) SAS
The triangles are congruent
Given: ΔABC ∼ ΔDEF
AB = 6, DE = 9, BC = 8, EF = 12, AC = 10, DF = 15
What theorem proves ΔABC ∼ ΔDEF?
A) AA
B) SSS
C) SAS
the answer is B) SSS
In ΔABC, ∠C = 90°, AB = 10, AC = 6. What is the length of side BC?
A) 6
B) 8
C) 12
the answer is B) 8
Find the surface area and volume of a cube with edge length 6 cm. What is the surface area of the cube?
A) 216 cm²
B) 180 cm²
C) 144 cm²
the answer is A)
Surface area of a cube = 6 × (edge length)²
= 6 × 6²
= 6 × 36
= 216 cm²
Volume of a cube = (edge length)³ = 6³ = 216 cm³
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is randomly selected, what's the probability it's blue?
A) 1/2
B) 1/5
C) 3/10
the answer is C)
Total marbles = 5 + 3 + 2 = 10 Blue marbles = 3
Probability = Blue marbles / Total marbles = 3/10
Given: ΔABC ≅ ΔDEF
∠A = ∠D = 60°
∠B = ∠E = 80°
AB = DE = 8
What theorem proves ΔABC ≅ ΔDEF?
A) SSS
B) SAS
C) ASA
D) AAS
The answer is C) ASA
This fits the ASA congruence theorem
Given: ΔABC ∼ ΔDEF
AB = 4, DE = 6, ∠A = ∠D = 50°, AC = 8, DF = 12
What theorem proves ΔABC ∼ ΔDEF?
A) AA
B) SSS
C) SAS
the answer is C) SAS because we hav two pairs of sides that seem to be proportional
AB/DE = 4/6= 2/3
AC/DF = 8/12= 2/3
In ΔABC, ∠C = 90°, AB = 15, ∠A = 30°. What is the length of side BC?
A) 7.5
B) 10
C) 12
the answer is A)
sin(∠A) = BC / AB sin(30°) = BC / 15 0.5 = BC / 15 BC = 15 × 0.5 BC = 7.5
A rectangular prism has dimensions:
Length = 8 cm
Width = 5 cm
Height = 3 cm
What is the volume of the prism?
A) 60 cm³
B) 100 cm³
C) 120 cm³
the answer is C)
Volume = Length × Width × Height = 8 × 5 × 3 = 120 cm³
Surface area = 2 × (Length × Width + Length × Height + Width × Height) = 2 × (8 × 5 + 8 × 3 + 5 × 3) = 2 × (40 + 24 + 15) = 2 × 79 = 158 cm²
A fair six-sided die is rolled. What's the probability of rolling an even number?
A) 1/3
B) 1/2
C) 2/3
the answer is B)
Even numbers: 2, 4, 6
Total numbers: 1, 2, 3, 4, 5, 6
Probability = Number of even numbers / Total numbers = 3/6 = 1/2
Given: ΔABC ≅ ΔDEF (right triangles)
AC = DF = 5 (hypotenuse)
BC = EF = 3 (leg)
What theorem proves ΔABC ≅ ΔDEF?
A) SSS
B) SAS
C) ASA
D) HL
The answer is D) HL
because we have right triangles with equal hypotenuses and equal legs, this also fits with the HL congruence theorem
Given: ΔABC ∼ ΔDEF
∠A = ∠D = 30°
∠B = ∠E = 60°
What theorem proves ΔABC ∼ ΔDEF?
A) AA
B) SSS
C) SAS
In ΔABC, ∠C = 90°, AC = 8, ∠A = 60°. What is the length of side AB?
A) 10
B) 12
C) 16
the answer is C)
cos(∠A) = AC / AB cos(60°) = 8 / AB 0.5 = 8 / AB AB = 8 / 0.5 AB = 16
A cylinder has:
Radius = 4 cm
Height = 10 cm
What is the volume of the cylinder?
A) 160π cm³
B) 120π cm³
C) 200π cm³
the answer is A)
Volume = π × Radius² × Height = π × 4² × 10 = π × 16 × 10 = 160π cm³
Surface area = 2 × π × Radius × (Radius + Height) = 2 × π × 4 × (4 + 10) = 2 × π × 4 × 14 = 112π cm²
A deck of 52 cards has 4 suits (hearts, diamonds, clubs, spades). What's the probability of drawing an ace?
A) 1/13
B) 1/52
C) 1/4
the answer is A)
Number of aces = 4
Total cards = 52
Probability = Number of aces / Total cards = 4/52 = 1/13
Given: ΔABC ≅ ΔDEF
AB = DE = 4
∠A = ∠D = 30°
AC = DF = 6
What theorem proves ΔABC ≅ ΔDEF?
A) SSS
B) SAS
C) ASA
D) AAS
The answer is B) SAS
we have two sides that are equal and this also fits the SAS congruence theorem
Given: ΔABC ∼ ΔDEF
AB = 3, DE = 6, BC = 4, EF = 8
What additional information would prove ΔABC ∼ ΔDEF by SAS?
A) ∠A = ∠D
B) ∠B = ∠E
C) ∠C = ∠F
the answer would be B) ∠B = ∠E because we have proportional sides such as
AB/DE = 3/6= 1/2
BC/EF = 4/8= 1/2
n ΔABC, ∠C = 90°, BC = 5, ∠A = 45°. What is the length of side AB?
A) 5√2
B) 5
C) 10
the answer is A)
sin(∠A) = BC / AB sin(45°) = 5 / AB 1/√2 = 5 / AB AB = 5√2
A sphere has radius = 6 cm.
What is the volume of the sphere?
A) 144π cm³
B) 216π cm³
C) 288π cm³
the answer is C)
Volume = (4/3) × π × Radius³ = (4/3) × π × 6³ = (4/3) × π × 216 = 288π cm³
Surface area = 4 × π × Radius² = 4 × π × 6² = 4 × π × 36 = 144π cm²
A bag contains 8 white balls and 4 black balls. If 2 balls are drawn randomly without replacement, what's the probability that both balls are white?
A) 8/33
B) 14/33
C) 2/3
the answer is A)
Probability of both balls being white = (8/12) × (7/11) = (2/3) × (7/11) = 14/33
Given: ΔABC ≅ ΔDEF
∠A = ∠D = 40°
∠B = ∠E = 60°
∠C = ∠F = 80°
What additional information would prove ΔABC ≅ ΔDEF? Any of the following would work AB = DE, BC = EF, or AC = DF
What theorem would prove ΔABC ≅ ΔDEF?
A) SSS
B) SAS
C) ASA
D) AAS
Given AB = DE, the answer is D) AAS
with ∠A = ∠D, ∠B = ∠E, and AB = DE, this suits the AAS congruence theorem
Given: ΔABC ∼ ΔXYZ
AB = 5, XY = 10, AC = 7, XZ = 14
Which theorem proves ΔABC ∼ ΔXYZ?
A) AA
B) SSS
C) SAS
The answer is C) SAS because sides :
AB/XY = 5/10= 1/2
aAC/XZ = 7/14= 1/2
are proportional and the including angles ∠A and ∠X would be equal
In ΔABC, ∠C = 90°, AB = 8, ∠B = 60°. What is the length of side AC?
A) 4
B) 4√3
C) 6
the answer is A)
sin(30°) = AC / 8 0.5 = AC / 8 AC = 4
A cone has:
Radius = 3 cm
Height = 7 cm
What is the volume of the cone?
A) 15π cm³
B) 21π cm³
C) 25π cm³
the answer is B)
Volume = (1/3) × π × Radius² × Height = (1/3) × π × 3² × 7 = (1/3) × π × 9 × 7 = 21π cm³
Slant height = √(Radius² + Height²)
\ = √(3² + 7²)
= √(9 + 49)
= √58
Surface area = π × Radius × (Radius + Slant height) = π × 3 × (3 + √58)
A coin is flipped 3 times. What's the probability of getting exactly 2 heads?
A) 1/8
B) 3/8
C) 1/2
the answer is B)
Probability = Favorable outcomes / Total outcomes = 3/8