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ISEE Upper Level Math : ISEE Upper Level (grades 9-12) Mathematics Achievement

Study concepts, example questions & explanations for ISEE Upper Level Math

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find The Length Of A Side In Other Polygons

In a septagon, each of the sides are equal to one another. If the perimeter is feet, what is the length of one of the sides?

Possible Answers:

15 ft

14 ft

12 ft

7 ft

Correct answer:

14 ft

Explanation:

If the perimeter of a septagon, in which each side is equal to the other, is 98, the length of each side will be 7 feet because 98 divided by 7 is 14.

$P=7\cdot s$

$98=7\cdot s$

$\frac{98}{7}=s$

s=14

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Example Question #1 : Lines

Find the distance between (1,4 ) and (-2,3) .

Possible Answers:

$\sqrt{10}$

$\sqrt{5}$

$\sqrt{2}$

Correct answer:

$\sqrt{10}$

Explanation:

To find the distance, first remember the distance formula: $\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$ . Plug in so that you have: $\sqrt{(1+2)^2+(4-3)^2}$ . Simplify so that you get $\sqrt{3^2+1^2}$ . This yields $\sqrt{10}$ .

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Example Question #1 : How To Find An Angle

Lines

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

Possible Answers:

165-x

131 -x

165 +x

75 + 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 #31 : Geometry

Lines

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

Possible Answers:

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

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

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

92 - 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 #31 : Isee Upper Level (Grades 9 12) Mathematics Achievement

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 #3 : Lines

Lines

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

Possible Answers:

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

$\angle 2 \cong \angle 6$

$\angle 5 \cong \angle 4$

$\angle 5 \cong \angle 8$

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

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 #251 : Geometry

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

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

What is $m \angle 3$ ?

Possible Answers:

$100^{\circ }$

$50^{\circ }$

$80^{\circ }$

$10^{\circ }$

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 #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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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : ISEE Upper Level (grades 9-12) Mathematics Achievement

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Find The Length Of A Side In Other Polygons

Example Question #1 : Lines

Example Question #1 : How To Find An Angle

Example Question #31 : Geometry

Example Question #31 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #3 : Lines

Example Question #251 : Geometry

Example Question #2 : How To Find An Angle

Example Question #1 : How To Find An Angle

Example Question #2041 : Hspt Mathematics

All ISEE Upper Level Math Resources

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