PARCC Geometry Practice Test 2 — Questions and Answers

Question 1: A triangle has angles measuring 47°, 68°, and x°. What is the value of x?

65° (Correct answer)
115°
75°
55°

Correct answer: 65°

47 + 68 + x = 180 → 115 + x = 180 → x = 65°.

The Triangle Angle Sum Theorem states that the interior angles of any triangle (in Euclidean geometry) always add up to 180°. This is a foundational PARCC geometry concept. Setting up the equation: 47° + 68° + x° = 180° 115° + x° = 180° x° = 180° − 115° = 65° The missing angle is 65°. Verification: 47 + 68 + 65 = 180 ✓ Classifying the triangle by angles: - All angles less than 90° → acute triangle - One angle equal to 90° → right triangle - One angle greater than 90° → obtuse triangle This triangle has angles 47°, 65°, and 68° — all less than 90°, so it is an ACUTE triangle. Extension concept: the exterior angle of a triangle equals the sum of the two non-adjacent interior angles. If an exterior angle at the third vertex were drawn, it would measure 180° − 65° = 115° = 47° + 68° ✓ Common mistakes: - Adding only two angles and forgetting that three angles must sum to 180°. - Computing 180 − 47 − 68 incorrectly due to arithmetic errors.

Question 2: Two parallel lines are cut by a transversal. One angle measures 73°. What is the measure of its co-interior (same-side interior) angle?

107° (Correct answer)
73°
17°
117°

Correct answer: 107°

Co-interior angles are supplementary: 73° + x = 180° → x = 107°.

When a transversal crosses two parallel lines, it creates several pairs of angle relationships. This is a fundamental PARCC geometry topic. Key angle relationships: - Corresponding angles: equal (same position at each intersection) - Alternate interior angles: equal (Z-shape or N-shape) - Alternate exterior angles: equal - Co-interior angles (same-side interior / consecutive interior): supplementary (add to 180°) For this problem, the angle is 73° and we need its co-interior (same-side interior) angle. Co-interior angles are supplementary: 73° + x = 180° x = 107° Why are co-interior angles supplementary? Both angles are on the same side of the transversal and between the parallel lines. Together they form a straight angle when the parallel lines' properties are applied. Common mistakes: - Confusing co-interior with alternate interior angles (which are equal, not supplementary). The 'same-side' part is the key — same side = supplementary. - Choosing 73° (alternate interior angle, which would be equal) or 17° (complement, which has no role here). Memory tip: Same Side → Supplementary (both start with S).

Question 3: A rectangle has a diagonal of 13 cm and a width of 5 cm. What is the length of the rectangle?

12 cm (Correct answer)
10 cm
8 cm
11 cm

Correct answer: 12 cm

5² + l² = 13² → 25 + l² = 169 → l² = 144 → l = 12 cm.

Applying the Pythagorean theorem to rectangles is a fundamental PARCC geometry skill. The diagonal of a rectangle divides it into two congruent right triangles, where the legs are the length and width, and the hypotenuse is the diagonal. Pythagorean Theorem: a² + b² = c² (where c is the hypotenuse) Here: - One leg (width) = 5 cm - Hypotenuse (diagonal) = 13 cm - Other leg (length) = ? 5² + l² = 13² 25 + l² = 169 l² = 169 − 25 = 144 l = √144 = 12 cm This is the 5-12-13 Pythagorean triple — a well-known combination that students should recognize on PARCC. Common Pythagorean triples to memorize: - 3-4-5 (and multiples: 6-8-10, 9-12-15) - 5-12-13 - 8-15-17 Area of the rectangle: length × width = 12 × 5 = 60 cm² Perimeter: 2(12) + 2(5) = 24 + 10 = 34 cm Common mistakes: - Adding legs instead of squaring: 5 + l = 13 → l = 8 (wrong — this ignores the squaring step). - Treating the diagonal as a leg and solving for the hypotenuse instead.

Question 4: What is the area of a trapezoid with parallel bases of 8 cm and 14 cm and a height of 6 cm?

66 cm² (Correct answer)
54 cm²
72 cm²
48 cm²

Correct answer: 66 cm²

A = ½ × (8 + 14) × 6 = ½ × 22 × 6 = 66 cm².

The area of a trapezoid is a standard PARCC geometry formula. Students must memorize and correctly apply: A = ½(b₁ + b₂)h Where: - b₁ = first parallel base = 8 cm - b₂ = second parallel base = 14 cm - h = height (perpendicular distance between the bases) = 6 cm Calculation: A = ½ × (8 + 14) × 6 A = ½ × 22 × 6 A = ½ × 132 A = 66 cm² Why this formula works: A trapezoid can be thought of as the average of its two parallel bases, times the height. This is equivalent to finding the area of a parallelogram with a base equal to the average of the two parallel sides. Alternative thinking: a trapezoid can be split into triangles and rectangles, and the areas summed — but the formula is faster. Common mistakes: - Forgetting the ½: computing (8 + 14) × 6 = 132 instead of 66. - Using the slant side as the height. The height must be PERPENDICULAR to both bases. - Adding only one base instead of both: ½ × 8 × 6 = 24 or ½ × 14 × 6 = 42.

Question 5: Circle O has a radius of 9 cm. What is the length of an arc that subtends a central angle of 80°?

4π cm (Correct answer)
16π cm
8π cm
2π cm

Correct answer: 4π cm

Arc length = (80/360) × 2π(9) = (2/9) × 18π = 4π cm.

Arc length is a foundational PARCC geometry concept involving circles. The formula connects the arc to the full circumference. Formula: Arc length = (θ/360°) × 2πr Where: - θ = central angle = 80° - r = radius = 9 cm Calculation: Arc length = (80/360) × 2π(9) = (80/360) × 18π = (2/9) × 18π = 36π/9 = 4π cm Numerically: 4π ≈ 4 × 3.14159 ≈ 12.57 cm Why the formula works: The full circumference corresponds to 360°. An arc subtending 80° is 80/360 = 2/9 of the full circle, so the arc length is 2/9 of the circumference. Related concept — sector area: A = (θ/360°) × πr² = (80/360) × π(81) = (2/9) × 81π = 18π cm² Common mistakes: - Using diameter instead of radius: substituting 18 for r gives the wrong answer. - Not simplifying 80/360 before multiplying, leading to arithmetic errors. - Confusing arc length (a linear measurement) with arc area (which isn't a standard measurement — sectors have area).

Question 6: Triangle ABC is similar to triangle DEF. If AB = 6, BC = 9, and DE = 10, what is the length of EF?

15 (Correct answer)
12
18
20

Correct answer: 15

Scale factor: DE/AB = 10/6 = 5/3. EF corresponds to BC: EF = BC × (5/3) = 9 × (5/3) = 15.

Similar triangles have proportional corresponding sides — a core PARCC geometry concept. Given: △ABC ~ △DEF This means the vertices correspond in order: A↔D, B↔E, C↔F Therefore the corresponding sides are: AB ↔ DE (given: AB = 6, DE = 10) BC ↔ EF (given: BC = 9, EF = ?) AC ↔ DF Step 1: Find the scale factor (ratio of similarity). Scale factor = DE/AB = 10/6 = 5/3 Step 2: Apply the scale factor to find EF. EF/BC = 5/3 EF = BC × (5/3) = 9 × (5/3) = 45/3 = 15 Alternative: set up a proportion. AB/DE = BC/EF 6/10 = 9/EF 6 × EF = 90 EF = 15 Verification: 6/10 = 9/15 → 0.6 = 0.6 ✓ Common mistakes: - Using the wrong pairs of corresponding sides. The similarity statement △ABC ~ △DEF tells you exactly which vertices match — always read the order carefully. - Inverting the ratio: using AB/DE = 6/10 when you need DE/AB = 10/6 (results in EF = 5.4, not 15). - Cross-multiplication errors.

PARCC Geometry Practice Test 2

A triangle has angles measuring 47°, 68°, and x°.
What is the value of x?