How to find an angle in a pentagon

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ISEE Upper Level Quantitative Reasoning › How to find an angle in a pentagon

Questions 1 - 4

The measures of the angles of a pentagon are $x ^{\circ }, x ^{\circ }, x ^{\circ }, y ^{\circ }, y ^{\circ }$ . If $x = 100 ^{\circ }$ , what is ?

$y = 120^{\circ}$

$y = 30^{\circ}$

$y = 80^{\circ}$

$y = 50^{\circ}$

This pentagon cannot exist.

Explanation

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know:

If $x = 100 ^{\circ }$ , then the above becomes:

$3 \cdot 100 + 2y = 540$

$2y \div 2 = 240 \div 2$

$y = 120^{\circ}$

The measures of the angles of a pentagon are $x^{\circ }, x^{\circ }, x^{\circ }, y^{\circ }, y^{\circ }$ . If , what is ?

$x = 113 \frac{1}{3}^{\circ}$

$x = 53 \frac{1}{3}^{\circ}$

$x = 170^{\circ}$

$x = 90^{\circ}$

This pentagon cannot exist.

Explanation

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know that:

If , then

$3x + 2 \cdot 100 = 540$

$x = 340 \div 3$

$x = 113 \frac{1}{3}^{\circ}$

Solve for :

Question_11

Explanation

The sum of the interior angles of a pentagon can be determined by the following equation, where is the number of sides:

Therefore:

A convex pentagon has four angles that measure $150^{\circ }$ each. What is the measure of the fifth angle?

This pentagon cannot exist.

$60^{\circ }$

$90^{\circ }$

$30^{\circ }$

$120^{\circ }$

Explanation

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . If we let the unknown angle measure be $x^{\circ }$ , then from this information:

Since an angle measure cannot be negative, this pentagon cannot exist.

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