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HSPT Math : Concepts

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Example Questions

Example Question #273 : Geometry

Two angles are complementary. The measure of one angle is 23 degrees less than twice that of the second. Give the greater of the measures of the two angles.

Possible Answers:

$52 \frac{1}{3} ^{\circ }$

$67 \frac{2}{3} ^{\circ }$

$78 \frac{1}{2}^{\circ }$

$37 \frac{2}{3}^{\circ }$

Correct answer:

$52 \frac{1}{3} ^{\circ }$

Explanation:

Let be the measure of one of the angles. The measure of the other angle is 23 degrees less than twice this, which is 2X - 23 . The angles are complementary, meaning that their measures add up to 90. Therefore:

(2X - 23) + X = 90

3X - 23 = 90

3X - 23+ 23 = 90 + 23

3X = 113

$3X\div 3 = 113 \div 3$

$X = 37 \frac{2}{3}$

The other angle has measure $90- 37 \frac{2}{3} = 52 \frac{1}{3}$ .

The greater of the two measures is requested, so the correct response is $52 \frac{1}{3} ^{\circ }$ .

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Example Question #1461 : Concepts

An angle supplementary to an angle of measure $137^{\circ }$ is complementary to an angle of what measure?

Possible Answers:

$37^{\circ }$

$68 \frac{1}{2} ^{\circ }$

$43 ^{\circ }$

$47^{\circ }$

Correct answer:

$47^{\circ }$

Explanation:

Two angles are supplementary if the sum of their degree measures is 180; Two angles are complementary if the sum of their degree measures is 90.

An angle supplementary to an angle with measure $137^{\circ }$ has measure

$180 ^{\circ } - 137^{\circ } = 43^{\circ }$ ;

this is complementary to an angle with measure

$90 ^{\circ } - 43^{\circ } = 47 ^{\circ }$ .

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Example Question #1462 : Concepts

An angle complementary to an angle of measure $24^{\circ }$ is supplementary to an angle of what measure?

Possible Answers:

$48^{\circ }$

$156 ^{\circ }$

$114 ^{\circ }$

$66 ^{\circ }$

Correct answer:

$114 ^{\circ }$

Explanation:

Two angles are supplementary if the sum of their degree measures is 180; Two angles are scomplementary if the sum of their degree measures is 90.

An angle complementary to an angle with measure $24^{\circ }$ has measure

$90 ^{\circ } - 24 ^{\circ } = 66 ^{\circ }$ ;

this is supplementary to an angle with measure

$180 ^{\circ } - 66 ^{\circ } = 114^{\circ }$ .

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Example Question #275 : Geometry

$\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$m \angle A = m \angle C = 60 ^{\circ }$ .

Is $\bigtriangleup DEF$ acute, right, or obtuse - or can it be determined?

Possible Answers:

$\bigtriangleup DEF$ is an acute triangle.

$\bigtriangleup DEF$ is a right triangle.

Whether $\bigtriangleup DEF$ is acute, right, or obtuse cannot be determined.

$\bigtriangleup DEF$ is an obtuse triangle.

Correct answer:

$\bigtriangleup DEF$ is an acute triangle.

Explanation:

Corresponding angles of similar triangles are congruent, so

$m \angle D = m \angle A = 60^{\circ }$

$m \angle F = m \angle C = 60^{\circ }$

The degree measures of the angles of a triangle total 180, so

$m \angle E =180^{\circ } - (m \angle D +m \angle F )$

$m \angle E =180^{\circ } - (60^{\circ } +60^{\circ } )$

$m \angle E =60^{\circ }$

All three angles of $\bigtriangleup DEF$ are acute, measuring less than 90 degrees, so $\bigtriangleup DEF$ is acute.

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Example Question #1465 : Concepts

$\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$m \angle A = m \angle C = 60 ^{\circ }$ .

Is $\bigtriangleup DEF$ scalene, isosceles, or equilateral - or can it be determined?

Possible Answers:

$\bigtriangleup DEF$ is a scalene triangle.

$\bigtriangleup DEF$ is a isosceles triangle, but not equilateral.

$\bigtriangleup DEF$ is an equilateral triangle.

Whether $\bigtriangleup DEF$ is scalene, isosceles, or equilateral cannot be determined.

Correct answer:

$\bigtriangleup DEF$ is an equilateral triangle.

Explanation:

Corresponding angles of similar triangles are congruent, so

$m \angle D = m \angle A = 60^{\circ }$

$m \angle F = m \angle C = 60^{\circ }$

The degree measures of the angles of a triangle total 180, so

$m \angle E =180^{\circ } - (m \angle D +m \angle F )$

$m \angle E =180^{\circ } - (60^{\circ } +60^{\circ } )$

$m \angle E =60^{\circ }$

The three angles of $\bigtriangleup DEF$ all measure the same, so it is equiangular; consequently, it is also equilateral.

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Example Question #281 : Geometry

In parallelogram ABCD , $m \angle A = (t+ 56 )^{\circ }$ . Give the measure of $\angle C$ in terms of .

Possible Answers:

$(t+ 146)^{\circ }$

$(124-t )^{\circ }$

$(t+ 56 )^{\circ }$

$(34-t )^{\circ }$

Correct answer:

$(t+ 56 )^{\circ }$

Explanation:

$\angle A$ and $\angle C$ are a pair of opposite angles of the parallelogram, and as such, they are congruent. Therefore,

$m \angle C = m \angle A = (t+ 56 )^{\circ }$ .

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Example Question #1467 : Concepts

The measures of the angles of $\bigtriangleup ABC$ are as follows:

$m \angle A = (t+35)^{\circ }$

$m \angle B = (3t-15)^{\circ }$

$m \angle C =(85-t)^{\circ }$

Is $\bigtriangleup ABC$ scalene, isosceles, or equilateral?

Possible Answers:

$\bigtriangleup ABC$ cannot exist

$\bigtriangleup ABC$ is equilateral

$\bigtriangleup ABC$ is isosceles but not equilateral

$\bigtriangleup ABC$ is scalene

Correct answer:

$\bigtriangleup ABC$ is equilateral

Explanation:

The sum of the measures of the angles of a triangle is 180 degrees, so solve for in the equation:

$m \angle A + m \angle B + m \angle C = 180^{\circ }$

$(t+35)^{\circ }+ (3t-15)^{\circ } + (85-t)^{\circ }= 180 ^{\circ }$

t+3t-t+35-15+85 = 180

3t +105= 180

3t +105- 105 = 180 - 105

3t =75

$3t \div 3 =75 \div 3$

t = 25

$m \angle A = (t+35)^{\circ } = (25+35)^{\circ } =60^{\circ }$

$m \angle B = (3t-15)^{\circ }= (3 \cdot 25-15)^{\circ } = (75-15)^{\circ } = 60^{\circ }$

$m \angle C =(85-t)^{\circ } =(85-25)^{\circ } = 60^{\circ}$

A triangle with three 60-degree angles is equilateral.

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Example Question #1468 : Concepts

$\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$m \angle A = m \angle C = 45 ^{\circ }$ .

Is $\bigtriangleup DEF$ acute, right, or obtuse - or can it be determined?

Possible Answers:

Whether $\bigtriangleup DEF$ is acute, right, or obtuse cannot be determined.

$\bigtriangleup DEF$ is a right triangle.

$\bigtriangleup DEF$ is an acute triangle.

$\bigtriangleup DEF$ is an obtuse triangle.

Correct answer:

$\bigtriangleup DEF$ is a right triangle.

Explanation:

The degree measures of the angles of a triangle total 180, so

$m \angle B =180^{\circ } - (m \angle A +m \angle C )$

$m \angle B =180^{\circ } - (45 ^{\circ } +45 ^{\circ } )$

$m \angle B =90^{\circ }$

Corresponding angles of similar triangles are congruent, so

$m \angle E = m \angle B = 90 ^{\circ }$

$\angle E$ is right, so $\bigtriangleup DEF$ is a right triangle.

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Example Question #1469 : Concepts

$m \angle A = 40^{\circ }$ in isosceles triangle $\bigtriangleup ABC$ . Is this triangle acute, right, or obtuse?

Possible Answers:

$\bigtriangleup ABC$ is acute

It is inconclusive whether $\bigtriangleup ABC$ is acute, right, or obtuse

$\bigtriangleup ABC$ is right

$\bigtriangleup ABC$ is obtuse

Correct answer:

It is inconclusive whether $\bigtriangleup ABC$ is acute, right, or obtuse

Explanation:

An isosceles triangle has at least two sides of equal measure, and also has at least two angles of equal measure.

One of two things can hold:

$m \angle A = 40^{\circ }$ , so either of $\angle B$ or $\angle C$ has measure $40^{\circ }$ . The third angle has measure

$[180 - (40+40 ) ]^{\circ }= (180-80) ^{\circ } = 100^{\circ }$ .

This angle, having measure greater than 90 degrees, is obtuse, so $\bigtriangleup ABC$ an obtuse triangle.

Alternatively, $m \angle B = \angle C = t ^{\circ }$ for some , so

40 + t + t = 40 + 2t = 180

40 + 2t - 40 = 180 - 40

2t = 140

$2t \div 2 = 140 \div 2$

t = 70

$m \angle B = m \angle C = 70 ^{\circ }$

Since all three angles have measure less than 90 degrees, all three angles are acute, and $\bigtriangleup ABC$ is an acute triangle.

Therefore, the information is inconclusive.

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Example Question #2061 : Hspt Mathematics

In parallelogram ABCD , $m \angle A = (t+ 56 )^{\circ }$ , and $m \angle B = 2t^{\circ }$ . Evaluate .

Possible Answers:

t= 56

$t = 41 \frac{1}{3}$

t = 62

t = 72

Correct answer:

$t = 41 \frac{1}{3}$

Explanation:

$\angle A$ and $\angle B$ are a pair of adjacent angles of the parallelogram, and as such, they are supplementary - that is, their degree measures total 180. Therefore,

$m \angle A + m \angle B = 180 ^{\circ }$

$(t+ 56)^{\circ } + 2t ^{\circ }= 180 ^{\circ }$

3t+56 = 180

3t+56 - 56 = 180 - 56

3t =124

$3t\div 3 =124 \div 3$

$t = 41 \frac{1}{3}$

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HSPT Math : Concepts

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #273 : Geometry

Example Question #1461 : Concepts

Example Question #1462 : Concepts

Example Question #275 : Geometry

Example Question #1465 : Concepts

Example Question #281 : Geometry

Example Question #1467 : Concepts

Example Question #1468 : Concepts

Example Question #1469 : Concepts

Example Question #2061 : Hspt Mathematics

All HSPT Math Resources

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