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

Possible Answers:

$26^{\circ}$

$36^{\circ}$

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

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

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

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 #1 : How To Find An Angle

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:

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

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

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

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

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

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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Example Question #2041 : Hspt Mathematics

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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

3x+14= 5x-17

14= 2x-17

2x= 31

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

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

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Example Question #1 : Lines

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

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

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

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

Evaluate x+y .

Possible Answers:

x+y = 60

x+y = 45

x+y = 30

x+y = 90

x+y = 120

Correct answer:

x+y = 60

Explanation:

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

3x+2y = 4x+21

3x+2y- 3x - 21 = 4x+21 - 3x - 21

x = 2y - 21

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

3x+2y + 2y-27 = 180

3x+4y-27 = 180

3x+4y= 207

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

3( 2y - 21)+4y= 207

6y-63+4y= 207

10y-63= 207

10y = 270

y = 27

Backsolve:

x = 2y - 21

x = 2 (27)- 21 = 54-21 = 33

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

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Example Question #2 : How To Find An Angle

Vertical_angles

Note: Figure NOT drawn to scale.

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

Possible Answers:

$88^{\circ }$

$144^{\circ }$

$96^{\circ }$

$120 ^{\circ }$

$128^{\circ }$

Correct answer:

$120 ^{\circ }$

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

2x+y = 2y ,

or, simplified,

2x+y - y= 2y - y

y = 2x

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

2y+ 8x = 180

Substitute for :

2(2x)+ 8x = 180

4x+8x = 180

12x= 180

x = 15

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

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Example Question #2 : Coordinate Geometry

Math4

What is the measurement of $\angle B$ ?

Possible Answers:

120

Correct answer:

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

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Example Question #41 : Triangles

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

Possible Answers:

$73^{\circ}$

$93^{\circ}$

$17^{\circ}$

$83^{\circ}$

Correct answer:

$83^{\circ}$

Explanation:

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

X+Y+Z=180

X+Y+97=180

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:

110

Correct answer:

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.

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.

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

The base angles must be degrees each.

As a check:

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

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Example Question #2041 : Hspt Mathematics

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

Possible Answers:

108

116

124

112

Correct answer:

108

Explanation:

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

$(n-2)\times 180$

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

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

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.

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

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Example Question #11 : Geometry

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

Possible Answers:

None of the other answers

$\dpi{100} \small 65$ degrees

$\dpi{100} \small 70$ degrees

$\dpi{100} \small 60$ degrees

$\dpi{100} \small 75$ degrees

Correct answer:

$\dpi{100} \small 60$ degrees

Explanation:

$\dpi{100} \small \angle AEC$ & $\dpi{100} \small \angle BEC$ must add up to 180 degrees. So, if $\dpi{100} \small \angle AEC$ is 120, $\dpi{100} \small \angle BEC$ (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 #1 : How To Find An Angle

Example Question #2041 : Hspt Mathematics

Example Question #1 : Lines

Example Question #2 : How To Find An Angle

Example Question #2 : Coordinate Geometry

Example Question #41 : Triangles

Example Question #262 : Geometry

Example Question #2041 : Hspt Mathematics

Example Question #11 : Geometry

All HSPT Math Resources

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