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HSPT Math : Geometry

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Example Questions

Example Question #2 : How To Find An Angle

Thingy

Note: Figure NOT drawn to scale.

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

Possible Answers:

$26^{\circ}$ $\displaystyle 26^{\circ}$

$36^{\circ}$ $\displaystyle 36^{\circ}$

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

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

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

Correct answer:

$36^{\circ}$ $\displaystyle 36^{\circ}$

Explanation:

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

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

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

$\displaystyle 8x+28 = 180$

8x=152 $\displaystyle 8x=152$

x = 19 $\displaystyle x = 19$

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

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Example Question #261 : Geometry

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

$\displaystyle (2x+72)+(5x-125)= 180$

$\displaystyle 7x - 53 = 180$

$\displaystyle 7x = 233$

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

The two angles have measure

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

and

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

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

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Example Question #2041 : Hspt Mathematics

Two vertical angles have measures $(3x+14)^{\circ }$ $\displaystyle (3x+14)^{\circ }$ and $\left ( 5x-17\right )^{\circ }$ $\displaystyle \left ( 5x-17\right )^{\circ }$. Which is the lesser of the measures (or the common measure) of the two angles?

Possible Answers:

$15\frac{1}{2}^{\circ }$ $\displaystyle 15\frac{1}{2}^{\circ }$

$60\frac{1}{2} ^{\circ }$ $\displaystyle 60\frac{1}{2} ^{\circ }$

$68\frac{5}{8}^{\circ}$ $\displaystyle 68\frac{5}{8}^{\circ}$

$22\frac{7}{8}^{\circ}$ $\displaystyle 22\frac{7}{8}^{\circ}$

$41\frac{1}{8}^{\circ}$ $\displaystyle 41\frac{1}{8}^{\circ}$

Correct answer:

$60\frac{1}{2} ^{\circ }$ $\displaystyle 60\frac{1}{2} ^{\circ }$

Explanation:

Two vertical angles - angles which share a vertex and whose union is a pair of lines - have the same measure. Therefore, we set up and solve the equation

$\displaystyle 3x+14= 5x-17$

$\displaystyle 14= 2x-17$

2x= 31 $\displaystyle 2x= 31$

$x= 15\frac{1}{2}$ $\displaystyle x= 15\frac{1}{2}$

$3x+14 = 3 \cdot 15\frac{1}{2} + 14 = 46\frac{1}{2} + 14= 60\frac{1}{2} ^{\circ }$ $\displaystyle 3x+14 = 3 \cdot 15\frac{1}{2} + 14 = 46\frac{1}{2} + 14= 60\frac{1}{2} ^{\circ }$

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Example Question #1 : Lines

A line $\displaystyle t$ intersects parallel lines $\displaystyle m$ and $\displaystyle n$. $\angle 1$ $\displaystyle \angle 1$ and $\angle 2$ $\displaystyle \angle 2$ are corresponding angles; $\angle 1$ $\displaystyle \angle 1$ and $\angle 3$ $\displaystyle \angle 3$ are same side interior angles.

$m \angle 1 = \left (3x+2y \right )^{\circ }$ $\displaystyle m \angle 1 = \left (3x+2y \right )^{\circ }$

$m \angle 2 = \left ( 4x+21\right )^{\circ }$ $\displaystyle m \angle 2 = \left ( 4x+21\right )^{\circ }$

$m \angle 3 = \left ( 2y-27\right )^{\circ}$ $\displaystyle m \angle 3 = \left ( 2y-27\right )^{\circ}$

Evaluate x+y $\displaystyle x+y$.

Possible Answers:

$\displaystyle x+y = 60$

$\displaystyle x+y = 45$

$\displaystyle x+y = 30$

$\displaystyle x+y = 90$

$\displaystyle x+y = 120$

Correct answer:

$\displaystyle x+y = 60$

Explanation:

When a transversal such as $\displaystyle t$ crosses two parallel lines, two corresponding angles - angles in the same relative position to their respective lines - are congruent. Therefore,

$\displaystyle 3x+2y = 4x+21$

$\displaystyle 3x+2y- 3x - 21 = 4x+21 - 3x - 21$

$\displaystyle x = 2y - 21$

Two same-side interior angles are supplementary - that is, their angle measures total 180 - so

$\displaystyle 3x+2y + 2y-27 = 180$

$\displaystyle 3x+4y-27 = 180$

$\displaystyle 3x+4y= 207$

We can solve this system by the substitution method as follows:

$\displaystyle 3( 2y - 21)+4y= 207$

$\displaystyle 6y-63+4y= 207$

$\displaystyle 10y-63= 207$

$\displaystyle 10y = 270$

y = 27 $\displaystyle y = 27$

Backsolve:

$\displaystyle x = 2y - 21$

$\displaystyle x = 2 (27)- 21 = 54-21 = 33$

$\displaystyle x+y = 27+33 = 60$, which is the correct response.

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Example Question #262 : Geometry

Vertical_angles

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the measure of $\angle 1$ $\displaystyle \angle 1$.

Possible Answers:

$120 ^{\circ }$ $\displaystyle 120 ^{\circ }$

$96^{\circ }$ $\displaystyle 96^{\circ }$

$128^{\circ }$ $\displaystyle 128^{\circ }$

$88^{\circ }$ $\displaystyle 88^{\circ }$

$144^{\circ }$ $\displaystyle 144^{\circ }$

Correct answer:

$120 ^{\circ }$ $\displaystyle 120 ^{\circ }$

Explanation:

The top and bottom angles, being vertical angles - angles which share a vertex and whose union is a pair of lines - have the same measure, so

$\displaystyle 2x+y = 2y$,

or, simplified,

$\displaystyle 2x+y - y= 2y - y$

y = 2x $\displaystyle y = 2x$

The right and bottom angles form a linear pair, so their degree measures total 180. That is,

$\displaystyle 2y+ 8x = 180$

Substitute $\displaystyle 2x$ for $\displaystyle y$:

$\displaystyle 2(2x)+ 8x = 180$

$\displaystyle 4x+8x = 180$

$\displaystyle 12x= 180$

x = 15 $\displaystyle x = 15$

The left and right angles, being vertical angles, have the same measure, so, since the right angle measures $\left (8x \right )^{\circ } =\left (8 \cdot 15 \right )^{\circ } = 120 ^{\circ }$ $\displaystyle \left (8x \right )^{\circ } =\left (8 \cdot 15 \right )^{\circ } = 120 ^{\circ }$, this is also the measure of the left angle, $\angle 1$ $\displaystyle \angle 1$.

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Example Question #2 : Coordinate Geometry

Math4

What is the measurement of $\angle B$ $\displaystyle \angle B$?

Possible Answers:

$\displaystyle 60$

$\displaystyle 90$

120 $\displaystyle 120$

$\displaystyle 80$

Correct answer:

$\displaystyle 60$

Explanation:

If you extend the lines of the parellelogram, you will notice that a parellogram is the same as 2 different sets of parellel lines intersecting one another. When that happens, the following angles are congruent to one another:

Math4-p1

Therefore, x = 60 $\displaystyle x = 60$

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Example Question #41 : Triangles

The three angles of a triangle are labeled $\displaystyle X$, $\displaystyle Y$, and $\displaystyle Z$. If $\displaystyle Z$ is $97^{\circ}$ $\displaystyle 97^{\circ}$, what is the value of X+Y $\displaystyle X+Y$?

Possible Answers:

$73^{\circ}$ $\displaystyle 73^{\circ}$

$93^{\circ}$ $\displaystyle 93^{\circ}$

$17^{\circ}$ $\displaystyle 17^{\circ}$

$83^{\circ}$ $\displaystyle 83^{\circ}$

Correct answer:

$83^{\circ}$ $\displaystyle 83^{\circ}$

Explanation:

Given that the three angles of a triangle always add up to 180 degrees, the following equation can be used:

$\displaystyle X+Y+Z=180$

$\displaystyle X+Y+97=180$

X+Y=83 $\displaystyle X+Y=83$

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Example Question #262 : Geometry

An isosceles triangle has a vertex angle of seventy degrees. What is the angle of one of the other angles?

Possible Answers:

$\displaystyle 35$

110 $\displaystyle 110$

$\displaystyle 70$

$\displaystyle 20$

$\displaystyle 55$

Correct answer:

$\displaystyle 55$

Explanation:

A triangle has three sides and three angles, which add up to 180 degrees.

An isosceles triangle must have 2 equal sides and 2 equal base angles. Given the vertex angle is 70 degrees, subtract this angle by 180.

$\displaystyle 180-70=110$

Since the other 2 base angles must equal to each other in an isoceles triangle, divide 110 with 2 to get the measure of the other angles.

$\frac{110}{2}=55$ $\displaystyle \frac{110}{2}=55$

The base angles must be $\displaystyle 55$ degrees each.

As a check:

$55+55+70=180 \textup{ degrees}$ $\displaystyle 55+55+70=180 \textup{ degrees}$

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Example Question #2041 : Hspt Mathematics

What is the measure of an interior angle of a regular pentagon?

Possible Answers:

108 $\displaystyle 108$

116 $\displaystyle 116$

124 $\displaystyle 124$

$\displaystyle 98$

112 $\displaystyle 112$

Correct answer:

108 $\displaystyle 108$

Explanation:

The formula to find the sum of total interior angles of a polygon is:

$(n-2)\times 180$ $\displaystyle (n-2)\times 180$

Since there are five sides in the pentagon, substitute n=5 $\displaystyle n=5$.

$(5-2)\times 180=540$ $\displaystyle (5-2)\times 180=540$

This is the sum of the interior angles of a pentagon. To find an interior angle, divide by five since there are five interior angles in a pentagon.

$\frac{540}{5}= 108$ $\displaystyle \frac{540}{5}= 108$

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Example Question #11 : Geometry

$\dpi{100} \small \overline{AB}$ $\displaystyle \dpi{100} \small \overline{AB}$ is a straight line. $\dpi{100} \small \overline{CD}$ $\displaystyle \dpi{100} \small \overline{CD}$ intersects $\dpi{100} \small \overline{AB}$ $\displaystyle \dpi{100} \small \overline{AB}$ at point $\dpi{100} \small E$ $\displaystyle \dpi{100} \small E$. If $\dpi{100} \small \angle AEC$ $\displaystyle \dpi{100} \small \angle AEC$ measures 120 degrees, what must be the measure of $\dpi{100} \small \angle BEC$ $\displaystyle \dpi{100} \small \angle BEC$?

Possible Answers:

None of the other answers

$\dpi{100} \small 65$ $\displaystyle \dpi{100} \small 65$ degrees

$\dpi{100} \small 70$ $\displaystyle \dpi{100} \small 70$ degrees

$\dpi{100} \small 60$ $\displaystyle \dpi{100} \small 60$ degrees

$\dpi{100} \small 75$ $\displaystyle \dpi{100} \small 75$ degrees

Correct answer:

$\dpi{100} \small 60$ $\displaystyle \dpi{100} \small 60$ degrees

Explanation:

$\dpi{100} \small \angle AEC$ $\displaystyle \dpi{100} \small \angle AEC$ & $\dpi{100} \small \angle BEC$ $\displaystyle \dpi{100} \small \angle BEC$ must add up to 180 degrees. So, if $\dpi{100} \small \angle AEC$ $\displaystyle \dpi{100} \small \angle AEC$ is 120, $\dpi{100} \small \angle BEC$ $\displaystyle \dpi{100} \small \angle BEC$ (the supplementary angle) must equal 60, for a total of 180.

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HSPT Math : Geometry

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #2 : How To Find An Angle

Example Question #261 : Geometry

Example Question #2041 : Hspt Mathematics

Example Question #1 : Lines

Example Question #262 : Geometry

Example Question #2 : Coordinate Geometry

Example Question #41 : Triangles

Example Question #262 : Geometry

Example Question #2041 : Hspt Mathematics

Example Question #11 : Geometry

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