HSPT Math : Geometry

Study concepts, example questions & explanations for HSPT Math

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

Possible Answers:

\(\displaystyle 26^{\circ}\)

\(\displaystyle 36^{\circ}\)

\(\displaystyle 7\frac{3}{4}^{\circ }\)

\(\displaystyle 29\frac{5}{7}^{\circ}\)

\(\displaystyle 21\frac{5}{7}^{\circ}\)

Correct answer:

\(\displaystyle 36^{\circ}\)

Explanation:

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

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

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

\(\displaystyle 8x+28 = 180\)

\(\displaystyle 8x=152\)

\(\displaystyle x = 19\)

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

Example Question #261 : Geometry

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

Possible Answers:

\(\displaystyle 33\frac{2}{7}^{\circ }\)

\(\displaystyle 41\frac{3}{7}^{\circ}\)

\(\displaystyle 22\frac{6}{7}^{\circ}\)

\(\displaystyle 65\frac{2}{3 } ^{\circ }\)

\(\displaystyle 81\frac{1}{2} ^{\circ }\)

Correct answer:

\(\displaystyle 41\frac{3}{7}^{\circ}\)

Explanation:

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

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

\(\displaystyle 7x - 53 = 180\)

\(\displaystyle 7x = 233\)

\(\displaystyle x = \frac{233}{7}= 33 \frac{2}{7}\)

The two angles have measure

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

and 

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

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

Example Question #2041 : Hspt Mathematics

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

Possible Answers:

\(\displaystyle 15\frac{1}{2}^{\circ }\)

\(\displaystyle 60\frac{1}{2} ^{\circ }\)

\(\displaystyle 68\frac{5}{8}^{\circ}\)

\(\displaystyle 22\frac{7}{8}^{\circ}\)

\(\displaystyle 41\frac{1}{8}^{\circ}\)

Correct answer:

\(\displaystyle 60\frac{1}{2} ^{\circ }\)

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

\(\displaystyle 3x+14= 5x-17\)

\(\displaystyle 14= 2x-17\)

\(\displaystyle 2x= 31\)

\(\displaystyle x= 15\frac{1}{2}\)

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

Example Question #1 : Lines

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

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

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

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

Evaluate \(\displaystyle x+y\).

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:

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

\(\displaystyle 6y-63+4y= 207\)

\(\displaystyle 10y-63= 207\)

\(\displaystyle 10y = 270\)

\(\displaystyle y = 27\)

Backsolve:

\(\displaystyle x = 2y - 21\)

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

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

Example Question #262 : Geometry

Vertical_angles

Note: Figure NOT drawn to scale.

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

Possible Answers:

\(\displaystyle 120 ^{\circ }\)

\(\displaystyle 96^{\circ }\)

\(\displaystyle 128^{\circ }\)

\(\displaystyle 88^{\circ }\)

\(\displaystyle 144^{\circ }\)

Correct answer:

\(\displaystyle 120 ^{\circ }\)

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,

\(\displaystyle 2x+y - y= 2y - y\)

\(\displaystyle y = 2x\)

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 \(\displaystyle y\):

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

\(\displaystyle 4x+8x = 180\)

\(\displaystyle 12x= 180\)

\(\displaystyle x = 15\)

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

Example Question #2 : Coordinate Geometry

Math4

What is the measurement of \(\displaystyle \angle B\)?

Possible Answers:

\(\displaystyle 60\)

\(\displaystyle 90\)

\(\displaystyle 120\)

\(\displaystyle 80\)

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, \(\displaystyle x = 60\)

Example Question #41 : Triangles

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

Possible Answers:

\(\displaystyle 73^{\circ}\)

\(\displaystyle 93^{\circ}\)

\(\displaystyle 17^{\circ}\)

\(\displaystyle 83^{\circ}\)

Correct answer:

\(\displaystyle 83^{\circ}\)

Explanation:

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

\(\displaystyle X+Y+Z=180\)

\(\displaystyle X+Y+97=180\)

\(\displaystyle X+Y=83\)

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:

\(\displaystyle 35\)

\(\displaystyle 110\)

\(\displaystyle 70\)

\(\displaystyle 20\)

\(\displaystyle 55\)

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.

\(\displaystyle \frac{110}{2}=55\)

The base angles must be \(\displaystyle 55\) degrees each.

As a check:

\(\displaystyle 55+55+70=180 \textup{ degrees}\)

Example Question #2041 : Hspt Mathematics

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

Possible Answers:

\(\displaystyle 108\)

\(\displaystyle 116\)

\(\displaystyle 124\)

\(\displaystyle 98\)

\(\displaystyle 112\)

Correct answer:

\(\displaystyle 108\)

Explanation:

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

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

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

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

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.

\(\displaystyle \frac{540}{5}= 108\)

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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