June 2022 Geometry Regents, Part I (multiple choice)

This exam was adminstered in June 2022.

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June 2022 Geometry Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


9. In the diagram below, lines k and ℓ intersect lines m and n at points A, B, C, and D.

Which statement is sufficient to prove ABCD is a parallelogram?

(1) ∠1 ≅ ∠3
(2) ∠4 ≅ ∠7
(3) ∠2 ≅ ∠5 and ∠5 ≅ ∠7
(4) ∠1 ≅ ∠3 and ∠3 ≅ ∠4

Answer: (3) ∠2 ≅ ∠5 and ∠5 ≅ ∠7

To prove that ABCD is a parallelogram, you need to show that AB || CD and AD || BC.

Choices (1) and (2) are aonly enough to show that one pair of lines are parallel, but not both.

In Choice (3), the first pair of angles shows that AB || CD and the second pair of angles shows AD || BC. This is the correct choice.

In Choice (4), ∠3 ≅ ∠4 is true because they are vertical angles. It doesn't both anything about the other line.

10. Which transformation does not always preserve distance?

(1) (x,y)→(x + 2, y)
(2) (x,y)→(-y, -x)
(3) (x,y)→(2x, y - 1)
(4) (x,y)→(3 - x, 2 - y)

Answer: (3) (x,y)→(2x, y - 1)

Dilations will not preserve distance. In Choice (3), the x coordinate is multiplied by 2.

Choice (1) is a translation, which preserves distance.

Choice (2) is a rotation of 180 degrees, followed by a reflection over y = x. Both of these preserve distance.

Choice (4) is a rotation and a translation. Both of these preserve distance.

11. In the diagram below, EF || HG, EF = 5, HG = 12, FI = 1.4x + 3, and HI = 6.1x - 6.5.

What is the length of HI?

(1) 1
(2) 5
(3) 10
(4) 24

Answer: (4) 24

The triangles are similar because EF || GH so the alternate interior angles are congruent as well as the vertical angles. This means that the corresponding sides are proportional.

(1.4x + 3) / 5 = (6.1x - 6.5) / 12

12(1.4x + 3) = 5(6.1x - 6.5)

16.8x + 36 = 30.5x - 32.5

36 = 13.7x - 32.5

68.5 = 13.7x

x = 5

If x = 5 then HI = 6.1(5) - 6.5 = 24, which is Choice (4).

12. The square pyramid below models a toy block made of maple wood.
Each side of the base measures 4.5 cm and the height of the pyramid is 10 cm. If the density of maple is 0.676 g/cm3, what is the mass of the block, to the nearest tenth of a gram?

(1) 45.6
(2) 67.5
(3) 136.9
(4) 202.5

Answer: (1) 45.6

Density is mass divided by the Volume, so m is Density times the Volume.

The Volume of the pyramid is 1/3 the Area of the base times the height.

V = 1/3 (4.5)² (10) = 67.5

m = d V = (0.676)(67.5) = 45.63

Choice (1) is the correct answer.

13. In the diagram below of right triangle EFG, altitude FH intersects hypotenuse EG at H.

If FH = 9 and EF = 15, what is EG?

(1) 6.75
(2) 12
(3) 18.75
(4) 25

Answer: (4) 25

The square of the altitude FH is equal to the product of EH and HG. You can find EH using the Pythagorean Theorem, if you didn't memorize the simpler Pythagorean Triples. (That is, multiples of 3-4-5, 5-12-13, etc.)

9² + EH² = 15²
81 + EH² = 225
EH² = 144
EH = 12

9² = 12 (HG)

81 = 12 HG

81/12 = HG

HG = 6.75

EG = EH + HG = 12 + 6.75 = 18.75

Choice (3) is the correct answer.

14. In triangle ABC below, D is a point on AB and E is a point on AC, such that DE || BC.

Which statement is always true?

(1) ∠ADE and ∠ABC are right angles.
(2) △ADE ∼ △ABC
(3) DE = 1/2 BC
(4) AD ≅ DB

Answer: (2) △ADE ∼ △ABC

If DE || BC, then the corresponding angles are congruent and the triangles are similar, which is Choice (2).

Choice (1) is incorrect because there is not that states that either angle is right or that AB is perpendicular to either DE or BC. You can't assum that it is right from the image.

Choices (3) and (4) would both be true if D was the midpoint of AB, but this is not stated as true. You cannot assume that it is the midpoint. Again, if D were the midpoint, then both choices would be true and that is not possible.

15. If one exterior angle of a triangle is acute, then the triangle must be

(1) right
(2) acute
(3) obtuse
(4) equiangular

Answer: (3) obtuse

If the triangle is equiangular, then it would also be acute, so Choice (4) makes no sense. There are three types of triangles, as classified by their angles, so the choice MUST be one of those three. The triangle cannot NOT be one of those three.

If the exterior angle is acute, then the interior angle is obtuse. So it is an obtuse triangle.

For the interior angle to be right, then the exterior angle would have to be right.

An acute triangle will have three obtuse exterior angles.

16. Given the information marked on the diagrams below, which pair of triangles can not always be proven congruent?

Answer: (4) [See Image]

Look for SSS, SAS, ASA, AAS or HL. If you see SSA (and it isn't HL), then that is the incorrect choice.

Choice (1) shows SAS, with right angles and the reflexive property.

Choice (2) shows AAS, with vertical angles being congruent.

Choice (3) shows SSS using the reflexive property.

Choice (4) shows SSA -- the congruent pair of angles is NOT between the two pairs of congruent sides. Note that it looks like they are both right triangles, but it is not stated if they are or are not.

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