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ISEE Upper Level Quantitative : Plane Geometry

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Example Questions

Example Question #81 : Plane Geometry

Which is the greater quantity?

(a) The measure of an angle complementary to a $55 ^{\circ }$ angle

(b) The measure of an angle supplementary to a $135 ^{\circ }$ angle

Possible Answers:

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

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

(a) The measure of an angle complementary to a $55 ^{\circ }$ angle is $90^{\circ } - 55^{\circ } = 35^{\circ }$

(b) The measure of an angle supplementary to a $135 ^{\circ }$ angle is $180^{\circ } - 135 ^{\circ } = 45^{\circ }$

This makes (b) greater.

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Example Question #2 : Lines

$\angle 1$ and $\angle 2$ are complementary; $m \angle 1 =2 m \angle 2 - 50$ .

Which is the greater quantity?

(A) $m \angle 1$

(B) $m \angle 2$

Possible Answers:

It is impossible to determine which is greater from the information given

(B) is greater

(A) is greater

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

Two angles are complementary if their degree measures total 90. Therefore,

$m \angle 1 + m \angle 2 = 90$

Since $m \angle 1 =2 m \angle 2 - 50$ , we can substitute, and we can solve for $m\angle2$ :

$(2 m \angle 2 - 50)+ m \angle 2 = 90$

$3 m \angle 2 - 50 = 90$

$3 m \angle 2 - 50 + 50 = 90 + 50$

$3 m \angle 2 = 140$

$3 m \angle 2 \div 3 = 140 \div 3$

$m \angle 2 = 46 \frac{2}{3}$

$m \angle 1 = 90 - m \angle 2 = 90 - 46 \frac{2}{3} = 43 \frac{1}{3}$

$m \angle 2 > m \angle 1$ , making (B) the greater quantity.

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Example Question #82 : Plane Geometry

Untitled

Note: diagram is not drawn to scale

Refer to the above diagram. If $m \angle1 = m \angle 2 + 30$ , what is $m \angle1$ ?

Possible Answers:

$100 ^{\circ }$

$110 ^{\circ }$

$105 ^{\circ }$

$120 ^{\circ }$

$115 ^{\circ }$

Correct answer:

$105 ^{\circ }$

Explanation:

$\angle 1$ and $\angle 2$ form a linear pair, so

$m \angle1 + m \angle 2 = 180$

Since $m \angle1 = m \angle 2 + 30$ , this can be rewritten as

$m \angle1 - 30 = m \angle 2$ , and the first equation can be rewritten as:

$m \angle1 + m \angle1 - 30 = 180$

$2 m \angle1 - 30 = 180$

$2 m \angle1 - 30 + 30 = 180 + 30$

$2 m \angle1 = 210$

$2 m \angle1 \div 2 = 210 \div 2$

$m \angle1 = 105$

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Example Question #83 : Plane Geometry

Untitled

Note: Figure NOT drawn to scale.

Which of the following pairs of numbers could give the measures of $\angle 1$ and $\angle 2$ ?

Possible Answers:

None of the pairs given in the other choices is correct.

$112^{\circ }, 68^{\circ }$

$64^{\circ }, 36^{\circ }$

$124^{\circ }, 76^{\circ }$

$55^{\circ }, 35^{\circ }$

Correct answer:

$112^{\circ }, 68^{\circ }$

Explanation:

The two angles form a linear pair and therefore their measures total $180^{\circ }$ . We check all of the pairs for this sum.

55+35 = 90

64+ 36 = 100

112+ 68 = 180

124+76 = 200

The correct pair is $112^{\circ }, 68^{\circ }$ .

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Example Question #5 : Lines

Triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which of the following triples could refer to the measures of $\angle 1$ , $\angle 2$ , and $\angle 3$ ?

Possible Answers:

$30 ^{\circ }, 80 ^{\circ }, 110 ^{\circ }$

All of the other responses are correct.

$25 ^{\circ }, 88 ^{\circ }, 113 ^{\circ }$

$42 ^{\circ }, 72 ^{\circ }, 114 ^{\circ }$

$56 ^{\circ }, 86 ^{\circ }, 142 ^{\circ }$

Correct answer:

All of the other responses are correct.

Explanation:

The measure of an exterior angle of a triangle, which here is $\angle 3$ , is the sum of the measures of its remote interior angles, which here are $\angle 1$ and $\angle 2$ . Therefore, we are looking for the sum of the first two angle measures to be equal to the third.

$42 ^{\circ }+ 72 ^{\circ } = 114 ^{\circ }$

$30 ^{\circ } + 80 ^{\circ } = 110 ^{\circ }$

$56 ^{\circ } + 86 ^{\circ } = 142 ^{\circ }$

$25 ^{\circ }+ 88 ^{\circ } = 113 ^{\circ }$

All four triples satisfy this condition.

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Example Question #7 : Lines

Untitled

Note: figure NOT drawn to scale

The degree measure of $\angle 1$ is five degrees greater than twice that of $\angle 2$ . Which is the greater quantity?

(A) $m \angle 2$

(B) $55^{\circ }$

Possible Answers:

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(A) is greater

(B) is greater

Correct answer:

(A) is greater

Explanation:

The degree measure of $\angle 1$ is five degrees greater than twice that of $\angle 2$ - that is,

$m \angle 1 = 2 m \angle 2 + 5$

Since $\angle 1$ and $\angle 2$ form a linear pair,

$m\angle1 + m\angle 2 = 180$ , and by substitution,

$( 2 m \angle 2 + 5)+ m\angle 2 = 180$

$3 m \angle 2 + 5 = 180$

$3 m \angle 2 + 5 - 5 = 180 - 5$

$3 m \angle 2 = 175$

$3 m \angle 2 \div 3 = 175\div 3$

$m \angle 2 = 58 \frac{1}{3} > 55$

This makes (A) greater.

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Example Question #8 : Lines

Untitled

Note: Figure NOT drawn to scale

$m\angle 2 = 48 ^{\circ }$ . What is $m \angle 1$ ?

Possible Answers:

$m \angle 1 = 142^{\circ }$

$m \angle 1 = 122^{\circ }$

$m \angle 1 = 112^{\circ }$

$m \angle 1 = 102^{\circ }$

$m \angle 1 = 132^{\circ }$

Correct answer:

$m \angle 1 = 132^{\circ }$

Explanation:

$\angle 1$ and $\angle 2$ form a linear pair and therfore, the the sum of their degree measures is $180^{\circ }$ .

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

$m\angle 1 + 48 ^{\circ } = 180^{\circ }$

$m\angle 1 + 48 ^{\circ } - 48 ^{\circ } = 180^{\circ } - 48 ^{\circ }$

$m\angle 1 = 132 ^{\circ }$

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Example Question #2 : Lines

Triangle

Note: Figure NOT drawn to scale.

$m \angle 3 = 125 ^{\circ }$

$3 m \angle 1 = 2 m \angle 2$

Evaluate $m \angle 1$ .

Possible Answers:

$m \angle 1 = 25^{\circ }$

$m \angle 1 = 60 ^{\circ }$

$m \angle 1 = 35 ^{\circ }$

$m \angle 1 = 50 ^{\circ }$

$m \angle 1 = 75 ^{\circ }$

Correct answer:

$m \angle 1 = 50 ^{\circ }$

Explanation:

$3 m \angle 1 = 2 m \angle 2$ , so

$3 \left (m \angle 1 \right ) \div 2 = 2\left ( m \angle 2 \right )\div 2$

$\frac{3}{2} m \angle 1 = m \angle 2$

The measure of an exterior angle of a triangle, which here is $\angle 3$ , is the sum of the measures of its remote interior angles, which here are $\angle 1$ and $\angle 2$ .

$m \angle 1 + m \angle 2 = m \angle 3$

$m \angle 1 + \frac{3}{2} m \angle 1 = 125 ^{\circ }$

$\frac{5}{2} m \angle 1 = 125 ^{\circ }$

$\frac{2}{5} \cdot \frac{5}{2} m \angle 1 = \frac{2}{5} \cdot 125 ^{\circ }$

$m \angle 1 =50 ^{\circ }$

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Example Question #81 : Plane Geometry

What is the slope of a line that passes through points (10,25) and (5, 10) ?

Possible Answers:

$\frac{1}{3}$

$-\frac{1}{3}$

Correct answer:

Explanation:

The equation for solving for the slope of a line is $m=\frac{y2-y1}{x2-x1}$ .

Thus, if (x1,y1)=(10, 25) and (x2,y2)=(5,10) , then:

$m=\frac{25-10}{10-5}=\frac{15}{5}=3$

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Example Question #81 : Plane Geometry

What is the slope of a line that passes through points (4.5,2.5) and (3.5, 6.5) ?

Possible Answers:

$-\frac{1}{4}$

$\frac{1}{4}$

Correct answer:

Explanation:

The equation for solving for the slope of a line is $m=\frac{y2-y1}{x2-x1}$ .

Thus, if (x1,y1)=(4.5, 2.5) and (x2,y2)=(3.5,6.5) , then:

$m=\frac{6.5-2.5}{3.5-4.5}=\frac{4}{-1}=-4$

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ISEE Upper Level Quantitative : Plane Geometry

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #81 : Plane Geometry

Example Question #2 : Lines

Example Question #82 : Plane Geometry

Example Question #83 : Plane Geometry

Example Question #5 : Lines

Example Question #7 : Lines

Example Question #8 : Lines

Example Question #2 : Lines

Example Question #81 : Plane Geometry

Example Question #81 : Plane Geometry

All ISEE Upper Level Quantitative Resources

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