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HSPT Math : HSPT Mathematics

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Example Questions

Example Question #1 : Geometry

Lines and are parallel. $\angle HFC=10^{\circ}$ , $\angle DGI=50^{\circ}$ , $\triangle EFG$ is a right triangle, and $\overline{EG}$ has a length of 10. What is the length of $\overline{EF}$

Possible Answers:

Not enough information.

Correct answer:

Explanation:

Since we know opposite angles are equal, it follows that angle $\angle AFE=10^{\circ}$ and $\angle BGE=50^{\circ}$ .

Imagine a parallel line passing through point . The imaginary line would make opposite angles with $\angle AFE$ & $\angle BGE$ , the sum of which would equal $\angle FEG$ . Therefore, $\angle FEG=60^{\circ}$ .

$\cos (60)=.5=\frac{EG}{EF}\rightarrow EF=\frac{EG}{.5}=20$

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Example Question #1 : Geometry

If $\angle A$ measures $(40-10x)^{\circ}$ , which of the following is equivalent to the measure of the supplement of $\angle A$ ?

Possible Answers:

$(10x+140)^{\circ}$

$(100x)^{\circ}$

$(10x+50)^{\circ}$

$(140-10x)^{\circ}$

$(50-10x)^{\circ}$

Correct answer:

$(10x+140)^{\circ}$

Explanation:

When the measure of an angle is added to the measure of its supplement, the result is always 180 degrees. Put differently, two angles are said to be supplementary if the sum of their measures is 180 degrees. For example, two angles whose measures are 50 degrees and 130 degrees are supplementary, because the sum of 50 and 130 degrees is 180 degrees. We can thus write the following equation:

$\dpi{100} measure\ of\ \angle A+ measure\ of\ supplement\ of\ \angle A=180$

$\dpi{100} 40-10x+ measure\ of\ supplement\ of\ \angle A=180$

Subtract 40 from both sides.

$\dpi{100} -10x+ measure\ of\ supplement\ of\ \angle A=140$

Add $\dpi{100} 10x$ to both sides.

$\dpi{100} measure\ of\ supplement\ of\ \angle A=140+10x=10x+140$

The answer is $(10x+140)^{\circ}$ .

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Example Question #2 : Geometry

In the following diagram, lines and are parallel to each other. What is the value for ?

Sat_math_166_03

Possible Answers:

100^o

It cannot be determined

60^o

30^o

80^o

Correct answer:

80^o

Explanation:

When two parallel lines are intersected by another line, the sum of the measures of the interior angles on the same side of the line is 180°. Therefore, the sum of the angle that is labeled as 100° and angle y is 180°. As a result, angle y is 80°.

Another property of two parallel lines that are intersected by a third line is that the corresponding angles are congruent. So, the measurement of angle x is equal to the measurement of angle y, which is 80°.

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Example Question #1 : Lines

Lines

Examine the above diagram. If $l \parallel m$ , give in terms of .

Possible Answers:

75 + x

131 -x

165-x

165 +x

75-x

Correct answer:

165-x

Explanation:

The two marked angles are same-side exterior angles of parallel lines, which are supplementary - that is, their measures have sum 180. We can solve for in this equation:

(y-17) + (x + 32) = 180

y + x-17 + 32= 180

y + x+15= 180

y + x+15- 15 - x = 180- 15 - x

y = 165- x

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Example Question #2 : Lines

Lines

Examine the above diagram. If $l \parallel m$ , give in terms of .

Possible Answers:

$92 - \frac{1}{2}x$

$47 - \frac{1}{2}x$

92 - x

$184 - \frac{1}{2}x$

47 - x

Correct answer:

$92 - \frac{1}{2}x$

Explanation:

The two marked angles are same-side interior angles of parallel lines, which are supplementary - that is, their measures have sum 180. We can solve for in this equation:

(2y + 15) + (x - 19) = 180

2y + x+ 15 - 19 = 180

2y + x-4 = 180

2y + x-4+4 -x = 180+4 -x

2y = 184 -x

$2y \div 2 = \left ( 184 -x \right )\div 2$

$y= 92 - \frac{1}{2}x$

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Example Question #33 : Plane Geometry

Lines

Examine the above diagram. What is x + y ?

Possible Answers:

112

Correct answer:

Explanation:

By angle addition,

$\left ( x - 13\right )+ 100 + (y + 25) = 180$

$x - 13\right+ 100 + y + 25 = 180$

$x + y - 13\right+ 100+ 25 = 180$

x + y +112 = 180

x + y +112 - 112= 180- 112

x + y = 68

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Example Question #1 : Lines

Lines

Examine the above diagram. Which of the following statements must be true whether or not and are parallel?

Possible Answers:

$\angle 2 \cong \angle 6$

$\angle 5 \cong \angle 4$

$\angle 5 \cong \angle 8$

$m \angle 4 + m \angle 7 = 180 ^{\circ }$

$m \angle 1 + m \angle 6 = 180 ^{\circ }$

Correct answer:

$\angle 5 \cong \angle 8$

Explanation:

Four statements can be eliminated by the various parallel theorems and postulates. Congruence of alternate interior angles or corresponding angles forces the lines to be parallel, so

$\angle 5 \cong \angle 4 \Rightarrow l \parallel m$ and

$\angle 2 \cong \angle 6 \Rightarrow l \parallel m$ .

Also, if same-side interior angles or same-side exterior angles are supplementary, the lines are parallel, so

$m \angle 4 + m \angle 7 = 180 ^{\circ } \Rightarrow l \parallel m$ and

$m \angle 1 + m \angle 6 = 180 ^{\circ } \Rightarrow l \parallel m$ .

However, $\angle 5 \cong \angle 8$ whether or not $l \parallel m$ since they are vertical angles, which are always congruent.

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Example Question #2 : Lines

$\angle 1$ and $\angle 2$ are supplementary; $\angle 1$ and $\angle 3$ are complementary.

$m \angle 2 = 100 ^{\circ }$ .

What is $m \angle 3$ ?

Possible Answers:

$100^{\circ }$

$10^{\circ }$

$80^{\circ }$

$50^{\circ }$

Correct answer:

$10^{\circ }$

Explanation:

Supplementary angles and complementary angles have measures totaling $180^{\circ }$ and $90^{\circ }$ , respectively.

$m \angle 2 = 100 ^{\circ }$ , so its supplement $\angle 1$ has measure

$m \angle 1 = 180^{\circ } - m \angle 2 = 180^{\circ } - 100 ^{\circ } = 80^{\circ }$

$\angle 3$ , the complement of $\angle 1$ , has measure

$m \angle 3 = 90^{\circ } - m \angle 1 = 90^{\circ } - 80 ^{\circ } = 10^{\circ }$

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Example Question #1 : Lines

Thingy

Note: Figure NOT drawn to scale.

In the above figure, $m \angle 1=\left ( x+17 \right )^{\circ}$ and $m \angle 2= \left (7x+11 \right )^{\circ}$ . Which of the following is equal to $m \angle 1$ ?

Possible Answers:

$29\frac{5}{7}^{\circ}$

$36^{\circ}$

$7\frac{3}{4}^{\circ }$

$26^{\circ}$

$21\frac{5}{7}^{\circ}$

Correct answer:

$36^{\circ}$

Explanation:

$\angle 1$ and $\angle 2$ form a linear pair, so their angle measures total $180^{\circ}$ . Set up and solve the following equation:

$m \angle 1+ m \angle 2 = 180^{\circ}$

$\left ( x+17 \right ) + \left (7x+11 \right ) = 180$

8x+28 = 180

8x=152

x = 19

$m \angle 1=\left ( x+17 \right )^{\circ} = \left ( 19+17 \right )^{\circ} = 36^{\circ}$

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Example Question #31 : Plane Geometry

Two angles which form a linear pair have measures $(2x+72)^{\circ}$ and $(5x-125)^{\circ}$ . Which is the lesser of the measures (or the common measure) of the two angles?

Possible Answers:

$33\frac{2}{7}^{\circ }$

$81\frac{1}{2} ^{\circ }$

$65\frac{2}{3 } ^{\circ }$

$22\frac{6}{7}^{\circ}$

$41\frac{3}{7}^{\circ}$

Correct answer:

$41\frac{3}{7}^{\circ}$

Explanation:

Two angles that form a linear pair are supplementary - that is, they have measures that total $180^{\circ}$ . Therefore, we set and solve for in this equation:

(2x+72)+(5x-125)= 180

7x - 53 = 180

7x = 233

$x = \frac{233}{7}= 33 \frac{2}{7}$

The two angles have measure

$2 \cdot 33 \frac{2}{7} +72= 138 \frac{4}{7} ^{\circ}$

and

$5 \cdot 33 \frac{2}{7} -125 = 41\frac{3}{7}^{\circ}$

$41\frac{3}{7}^{\circ}$ is the lesser of the two measures and is the correct choice.

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HSPT Math : HSPT Mathematics

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #1 : Geometry

Example Question #1 : Geometry

Example Question #2 : Geometry

Example Question #1 : Lines

Example Question #2 : Lines

Example Question #33 : Plane Geometry

Example Question #1 : Lines

Example Question #2 : Lines

Example Question #1 : Lines

Example Question #31 : Plane Geometry

All HSPT Math Resources

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