Rationale: Successive differences double each step: \(+2, +4, +8, +16\). The next difference is \(+32\); thus, \(33 + 32 = 65\).
6. Correct: D (49) G.GF.2
Rationale: The sequence consists of consecutive perfect squares: \(2^2=4, 3^2=9, 4^2=16, 5^2=25, 6^2=36\). Next is \(7^2 = 49\).
7. Correct: A G.GF.2
Contrapositive (\(\sim Q \to \sim P\)): Negate and swap both clauses: "If a polygon does not have six sides, then it is not a hexagon."
8. Correct: C G.GF.2
Converse (\(Q \to P\)): Switch hypothesis and conclusion: "If an angle is a right angle, then it measures \(90^\circ\)."
9. Correct: D G.GF.2
Inverse (\(\sim P \to \sim Q\)): Negate both clauses in original order: "If two lines are not parallel, then they intersect."
10. Correct: B ("They do not intersect") G.GF.2
Conclusion: The clause following "then" without the word "then" itself is the conclusion: "They do not intersect."
Page 1 of 2 • Geometry Benchmark Teacher Answer Key
Detailed algebraic steps, coordinate formulas, and common student errors
Questions 11–20
Part III: Points, Lines & Angles (Standard G.GF.1 • Questions 11–13)
11. Correct: C Point B
\(\angle 4\) is interior to \(\triangle ABC\) with rays \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\). The common endpoint (vertex) is Point B.
12. Correct: D A, B, and D
Points \(A\), \(B\), and \(D\) all lie directly along the single horizontal extended baseline line \(\overleftrightarrow{AD}\).
13. Correct: A \(\angle 1\) and \(\angle 3\)
Both \(\angle 1\) and \(\angle 3\) are visually marked with single blue congruence arcs and matching tick marks.
Part IV: Segment Addition Postulate & Algebra (Standard G.GF.1 • Questions 14–16)
14. Value of \(x\) B (\(x = 6\))
\(DE + EF = DF\)
\((4x - 1) + (2x + 7) = 8x - 6\)
\(6x + 6 = 8x - 6\)
\(12 = 2x \implies \mathbf{x = 6}\)
Common Pitfall: Adding \(DF\) instead of equating the sum of parts to whole.
15. Length of \(\overline{DF}\) D (\(DF = 42\))
Substitute \(x = 6\) into \(DF\):
\(DF = 8(6) - 6 = 48 - 6\)
\(\mathbf{DF = 42}\)
Check: \(23 + 19 = 42\) ✓
Distractor Alert: Choice A (23) is \(DE\); Choice B (19) is \(EF\).
16. Length of \(\overline{LM}\) C (\(LM = 45\))
\((7x + 3) + (3x - 1) = 12x - 10\)
\(10x + 2 = 12x - 10 \implies x = 6\)
\(LM = 7(6) + 3 = 42 + 3\)
\(\mathbf{LM = 45}\)
Distractor Alert: Choice A (6) is \(x\); Choice B (17) is \(MN\); Choice D (62) is \(LN\).
Part V: Coordinate Geometry (Midpoint & Distance) (Standard G.GF.7 • Questions 17–20)
17. Midpoint of \(\overline{GH}\) [G(7, 12), H(3, -4)] A \((5, 4)\)
\(M = \left(\frac{7 + 3}{2}, \frac{12 + (-4)}{2}\right) = \left(\frac{10}{2}, \frac{8}{2}\right) = \mathbf{(5, 4)}\)
Distractor B (2, 8): Subtracted coordinates instead of averaging them.
18. Distance of \(\overline{GH}\) [G(7, 12), H(3, -4)] D (\(16.5\))
\(d = \sqrt{(3 - 7)^2 + (-4 - 12)^2} = \sqrt{(-4)^2 + (-16)^2}\)
\(d = \sqrt{16 + 256} = \sqrt{272} \approx \mathbf{16.5}\)
Distractor C (20.0): Added raw deltas \(4 + 16\) without squaring and taking root.
19. Midpoint \(Q\) of \(\overline{PR}\) [P(4, -3), R(12, 5)] B \((8, 1)\)
\(Q = \left(\frac{4 + 12}{2}, \frac{-3 + 5}{2}\right) = \left(\frac{16}{2}, \frac{2}{2}\right) = \mathbf{(8, 1)}\)
Distractor A (16, 2): Added coordinates but forgot to divide each sum by 2.
20. Distance between J(-3, 6) and K(-7, 3) A (\(5.0\))
\(d = \sqrt{(-7 - (-3))^2 + (3 - 6)^2} = \sqrt{(-4)^2 + (-3)^2}\)
\(d = \sqrt{16 + 9} = \sqrt{25} = \mathbf{5.0}\)
Pythagorean Triple: Recognizable 3-4-5 right triangle relationship.
Grading Scale & Performance Bands
Exceeds (18–20 pts) 90%–100% • Advanced
Proficient (14–17 pts) 70%–85% • Target
Approaching (11–13 pts) 55%–65% • Review
Needs Support (≤10 pts) <55% • Intervention
Page 2 of 2 • Geometry Benchmark Teacher Answer Key • End of Document
A. \(\overline{QR} \cong \overline{QR}\)
Reason: Reflexive Property
B. \(PQ + QR = PR\)
Reason: Segment Addition Post.
C. \(\overline{PR} \cong \overline{QS}\)
Reason: Substitution Property
D. \(\overline{PQ} \cong \overline{RS}\)
Reason: Reflexive Property
Page 1 of 5 • Geometry Benchmark Study Guide
Standards: G.GF.2 • Identify numeric sequences and analyze conditional statements
Questions 5–10
Conditional
\(P \to Q\)
If P, then Q
Converse
\(Q \to P\)
If Q, then P
Inverse
\(\sim P \to \sim Q\)
If not P, then not Q
Contrapositive
\(\sim Q \to \sim P\)
If not Q, then not P
5. Find the next term in the sequence: G.GF.2
2, 6, 14, 30, 62, ______
A. 94
B. 126
C. 124
D. 128
6. Find the next number in the sequence: G.GF.2
1, 8, 27, 64, 125, ______
A. 196
B. 216
C. 225
D. 250
7. Given: "If a triangle is equilateral, then it is equiangular." G.GF.2
Which choice represents the contrapositive of the given statement?
A. If a triangle is not equiangular, then it is not equilateral.
B. If a triangle is equiangular, then it is equilateral.
C. If a triangle is not equilateral, then it is not equiangular.
D. A triangle is equilateral if and only if it is equiangular.
8. Given: "If two angles form a linear pair, then they are supplementary." G.GF.2
Which choice represents the converse of the given statement?
A. If two angles are not supplementary, then they do not form a linear pair.
B. If two angles are supplementary, then they form a linear pair.
C. If two angles do not form a linear pair, then they are not supplementary.
D. Two angles form a linear pair if and only if they are supplementary.
9. Given: "If a quadrilateral is a square, then it has four right angles." G.GF.2
Which choice represents the inverse of the given statement?
A. If a quadrilateral does not have four right angles, then it is not a square.
B. If a quadrilateral is not a square, then it does not have four right angles.
C. If a quadrilateral has four right angles, then it is a square.
D. A quadrilateral is a square if and only if it has four right angles.
10. Given: "If two coplanar lines do not intersect, then they are parallel." G.GF.2
Identify the hypothesis of the conditional statement.
A. Two coplanar lines do not intersect.
B. They are parallel.
C. If two coplanar lines do not intersect.
D. Lines are parallel.
Page 2 of 5 • Geometry Benchmark Study Guide
Standards: G.GF.1 • Diagram interpretation, collinear points, and angle identification
Questions 11–13
Use the diagram below for Questions 11, 12, and 13:
X Y W Z T 1 2 3 4 5
G.GF.1
A. Point Z
B. Point X
C. Point Y
D. Point T
G.GF.1
A. X, Y, and W
B. X, Y, and Z
C. Z, Y, and W
D. Y, T, and X
G.GF.1
A. \(\angle 1\) and \(\angle 3\)
B. \(\angle 4\) and \(\angle 5\)
C. \(\angle 2\) and \(\angle 3\)
D. \(\angle 1\) and \(\angle 4\)
Page 3 of 5 • Geometry Benchmark Study Guide
Standard: G.GF.1 • Apply algebraic reasoning to solve collinear segment lengths
Questions 14–16
14. Points \(A\), \(B\), and \(C\) all lie on the same line. Point \(B\) is between \(A\) and \(C\). G.GF.1
Given \(AB = 3x + 2\), \(BC = 2x + 8\), and \(AC = 7x - 4\). Find the value of \(x\).
A. \(x = 5\)
B. \(x = 7\)
C. \(x = 9\)
D. \(x = 12\)
Show your work:
15. Using the information from Question 14: G.GF.1
Given \(AB = 3x + 2\), \(BC = 2x + 8\), and \(AC = 7x - 4\). Find the length of segment \(\overline{AC}\).
A. \(AC = 23\)
B. \(AC = 45\)
C. \(AC = 22\)
D. \(AC = 51\)
Show your work:
16. Points \(P\), \(Q\), and \(R\) all lie on the same line. Point \(Q\) is between \(P\) and \(R\). G.GF.1
Given \(PQ = 5x + 1\), \(QR = 4x - 3\), and \(PR = 11x - 12\). Find the length of segment \(\overline{PQ}\).
A. \(PQ = 5\)
B. \(PQ = 17\)
C. \(PQ = 26\)
D. \(PQ = 43\)
Show your work:
Page 4 of 5 • Geometry Benchmark Study Guide
Standards: G.GF.7 • Midpoint formula and distance formula on the coordinate plane
Questions 17–20
Midpoint Formula Average Coordinates
\(M = \left(\frac{x_1 + x_2}{2}, \;\frac{y_1 + y_2}{2}\right)\)
Distance Formula Length Between Points
\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
17. Find the midpoint of segment \(\overline{AB}\) with endpoints \(A(6, 14)\) and \(B(2, -2)\). G.GF.7
Use the midpoint formula. Round any decimals to the nearest tenth if necessary.
A. \((4, 6)\)
B. \((2, 8)\)
C. \((4, 8)\)
D. \((8, 12)\)
18. Find the distance between points \(C(3, 9)\) and \(D(-3, 1)\). G.GF.7
Use the distance formula. Express your answer as an exact value or rounded to the nearest tenth.
A. \(8.0\)
B. \(10.0\)
C. \(14.0\)
D. \(12.5\)
19. Point \(S\) has coordinates \((1, -5)\) and point \(U\) has coordinates \((9, 7)\). G.GF.7
If point \(T\) is the midpoint of segment \(\overline{SU}\), what are the coordinates of point \(T\)?
A. \((10, 2)\)
B. \((5, 1)\)
C. \((5, 6)\)
D. \((4, 2)\)
20. Find the distance between point \(E(1, 2)\) and point \(F(6, 14)\). G.GF.7
Round all decimals to the nearest tenth if necessary.
A. \(13.0\)
B. \(15.2\)
C. \(17.0\)
D. \(11.8\)
Page 5 of 5 • Geometry Benchmark Study Guide • End of Practice
