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ISEE Upper Level Quantitative : Other Quadrilaterals

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Example Questions

Example Question #1 : Other Quadrilaterals

Three of the interior angles of a quadrilateral measure $100 ^{\circ }$ $\displaystyle 100 ^{\circ }$, $105^{\circ }$ $\displaystyle 105^{\circ }$, and $110 ^{\circ }$ $\displaystyle 110 ^{\circ }$. What is the measure of the fourth interior angle?

Possible Answers:

This quadrilateral cannot exist.

$15 ^{\circ }$ $\displaystyle 15 ^{\circ }$

$35^{\circ }$ $\displaystyle 35^{\circ }$

$45^{\circ }$ $\displaystyle 45^{\circ }$

$25^{\circ }$ $\displaystyle 25^{\circ }$

Correct answer:

$45^{\circ }$ $\displaystyle 45^{\circ }$

Explanation:

The measures of the angles of a quadrilateral have sum $360^{\circ }$ $\displaystyle 360^{\circ }$. If $\displaystyle x$ is the measure of the unknown angle, then:

$\displaystyle x + 100 + 105 + 110 = 360$

$\displaystyle x + 315= 360$

$\displaystyle x + 315-315= 360-315$

x = 45 $\displaystyle x = 45$

The measure of the fourth angle is $45^{\circ }$ $\displaystyle 45^{\circ }$.

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Example Question #2 : Quadrilaterals

In a certain quadrilateral, three of the angles are $65^{\circ}$ $\displaystyle 65^{\circ}$, $120^{\circ}$ $\displaystyle 120^{\circ}$, and $34^{\circ}$ $\displaystyle 34^{\circ}$. What is the measure of the fourth angle?

Possible Answers:

$29^{\circ}$ $\displaystyle 29^{\circ}$

$219^{\circ}$ $\displaystyle 219^{\circ}$

$141^{\circ}$ $\displaystyle 141^{\circ}$

$139^{\circ}$ $\displaystyle 139^{\circ}$

$41^{\circ}$ $\displaystyle 41^{\circ}$

Correct answer:

$141^{\circ}$ $\displaystyle 141^{\circ}$

Explanation:

A quadrilateral has four angles totalling $360^{\circ}$ $\displaystyle 360^{\circ}$. So, first add up the three angles given. The sum is $219^{\circ}$ $\displaystyle 219^{\circ}$. Then, subtract that from 360. This gives you the missing angle, which is $141^{\circ}$ $\displaystyle 141^{\circ}$.

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Example Question #201 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

The angles of Quadrilateral A measure $80^{\circ }, 80^{\circ }, 80^{\circ }, x^{\circ }$ $\displaystyle 80^{\circ }, 80^{\circ }, 80^{\circ }, x^{\circ }$

The angles of Pentagon B measure $100^{\circ }, 100^{\circ }, 100^{\circ }, 100^{\circ }, y^{\circ }$ $\displaystyle 100^{\circ }, 100^{\circ }, 100^{\circ }, 100^{\circ }, y^{\circ }$

Which is the greater quantity?

(A) $\displaystyle x$

(B) $\displaystyle y$

Possible Answers:

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

(A) and (B) are equal

(B) is greater

(A) is greater

Correct answer:

(B) is greater

Explanation:

The sum of the measures of the angles of a quadrilateral is $360^{\circ }$ $\displaystyle 360^{\circ }$; the sum of the measures if a pentagon is $540^{\circ }$ $\displaystyle 540^{\circ }$. Therefore,

$\displaystyle x + 80 + 80 + 80 = 360$

$\displaystyle x + 240 = 360$

x = 120 $\displaystyle x = 120$

and

$\displaystyle y + 100+ 100+ 100+ 100 = 540$

$\displaystyle y + 4 00 = 540$

y = 140 $\displaystyle y = 140$

y > x $\displaystyle y > x$, so (B) is greater.

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Example Question #1 : Other Quadrilaterals

Possible Answers:

$35^{\circ }$ $\displaystyle 35^{\circ }$

$25^{\circ }$ $\displaystyle 25^{\circ }$

$15 ^{\circ }$ $\displaystyle 15 ^{\circ }$

$45^{\circ }$ $\displaystyle 45^{\circ }$

This quadrilateral cannot exist.

Correct answer:

$45^{\circ }$ $\displaystyle 45^{\circ }$

Explanation:

The measures of the angles of a quadrilateral have sum $360^{\circ }$ $\displaystyle 360^{\circ }$. If $\displaystyle x$ is the measure of the unknown angle, then:

$\displaystyle x + 100 + 105 + 110 = 360$

$\displaystyle x + 315= 360$

$\displaystyle x + 315-315= 360-315$

x = 45 $\displaystyle x = 45$

The measure of the fourth angle is $45^{\circ }$ $\displaystyle 45^{\circ }$.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Other Quadrilaterals

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Other Quadrilaterals

Example Question #2 : Quadrilaterals

Example Question #201 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #1 : Other Quadrilaterals

All ISEE Upper Level Quantitative Resources

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