How to find an angle in a polygon - PSAT Math

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Question

A regular polygon has a measure of $140^\circ$ for each of its internal angles. How many sides does it have?

Answer

To determine the measure of the angles of a regular polygon use:

Angle = (n – 2) x 180° / n

Thus, (n – 2) x 180° / n = 140°

180° n - 360° = 140° n

40° n = 360°

n = 360° / 40° = 9

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Question

A regular seven sided polygon has a side length of 14”. What is the measurement of one of the interior angles of the polygon?

Answer

The formula for of interior angles based on a polygon with a number of side n is:

Each Interior Angle = (n-2)180/n

= (7-2)180/7 = 128.57 degrees

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Question

If angle A and angle C are complementary angles and B and D are supplementary angles, which of the following must be true?

Answer

This question is very misleading, because while each answer COULD be true, none of them MUST be true. Between angle A and C, onne of the angles could be very small (0.001 degrees) and the other one could be very large. For instance, if A = 89.9999 and C = 0.0001, AC = 0.009. On the other hand, the two angles could be very siimilar. If B = 90 and D = 90 then BD = 8100 and BD > AC. If we use these same values we disprove AD = BC as 8100 ≠ .009. Finally, if B is a very small value, then B/C will be very small and smaller than A/D.

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Question

In isosceles triangle ABC, the measure of angle A is 50 degrees. Which is NOT a possible measure for angle B?

Answer

If angle A is one of the base angles, then the other base angle must measure 50 degrees. Since 50 + 50 + x = 180 means x = 80, the vertex angle must measure 80 degrees.

If angle A is the vertex angle, the two base angles must be equal. Since 50 + x + x = 180 means x = 65, the two base angles must measure 65 degrees.

The only number given that is not possible is 95 degrees.

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Question

In triangle ABC, the measure of angle A = 70 degrees, the measure of angle B = x degrees, and the measure of angle C = y degrees. What is the value of y in terms of x?

Answer

Since the three angles of a triangle sum to 180, we know that 70 + x + y = 180. Subtract 70 from both sides and see that x + y = 110. Subtract x from both sides and see that y = 110 – x.

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Question

What is the measure, in degrees, of each interior angle of a regular convex polygon that has twelve sides?

Answer

The sum of the interior angles, in degrees, of a regular polygon is given by the formula 180(n – 2), where n is the number of sides. The problem concerns a polygon with twelve sides, so we will let n = 12. The sum of the interior angles in this polygon would be 180(12 – 2) = 180(10) = 1800.

Because the polygon is regular (meaning its sides are all congruent), all of the angles have the same measure. Thus, if we divide the sum of the measures of the angles by the number of sides, we will have the measure of each interior angle. In short, we need to divide 1800 by 12, which gives us 150.

The answer is 150.

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Question

In the figure above, polygon ABDFHGEC is a regular octagon. What is the measure, in degrees, of angle FHI?

Answer

Angle FHI is the supplement of angle FHG, which is an interior angle in the octagon. When two angles are supplementary, their sum is equal to 180 degrees. If we can find the measure of each interior angle in the octagon, then we can find the supplement of angle FHG, which will give us the measure of angle FHI.

The sum of the interior angles in a regular polygon is given by the formula 180(n – 2), where n is the number of sides in the polygon. An octagon has eight sides, so the sum of the angles of the octagon is 180(8 – 2) = 180(6) = 1080 degrees. Because the octagon is regular, all of its sides and angles are congruent. Thus, the measure of each angle is equal to the sum of its angles divided by 8. Therefore, each angle in the polygon has a measure of 1080/8 = 135 degrees. This means that angle FHG has a measure of 135 degrees.

Now that we know the measure of angle FHG, we can find the measure of FHI. The sum of the measures of FHG and FHI must be 180 degrees, because the two angles form a line and are supplementary. We can write the following equation:

Measure of FHG + measure of FHI = 180

135 + measure of FHI = 180

Subtract 135 from both sides.

Measure of FHI = 45 degrees.

The answer is 45.

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Question

What is the measure of each angle in a regular octagon?

Answer

An octagon contains six triangles, or 1080 degrees. This means with 8 angles, each angle is 135 degrees.

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Question

What is the measure of each central angle of an octagon?

Answer

There are 360 degrees and 8 angles, so dividing leaves 45 degrees per angle.

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure. $\Delta OKR$ is equilateral and Pentagon is regular.

Evaluate $m \angle MNR$ .

Answer

By angle addition,

$m \angle MNR + m \angle RNO = m \angle MNO$

$\angle MNO$ is an angle of a reguar pentagon, so its measure is $\frac{180^{\circ}\times (5-2)}{5} = 108 ^{\circ}$ .

To find $m \angle RNO$ , first we find $m \angle NOR$ .

By angle addition,

$m \angle NOR + m \angle ROK = m \angle NOK$

$\angle NOK$ is an angle of a regular pentagon and has measure $108 ^{\circ}$ .

$\angle ROK$ , as an angle of an equilateral triangle, has measure $60 ^{\circ }$ .

$m \angle NOR + 60 ^{\circ }= 108 ^{\circ }$

$m \angle NOR = 48 ^{\circ }$

$\Delta OKR$ is equilateral, so $\overline{OR} \cong \overline{OK}$ ; Pentagon is regular, so $\overline{OK} \cong \overline{NO}$ . Therefore, $\overline{OR} \cong \overline{ON}$ , and by the Isosceles Triangle Theorem, $m \angle NRO = m \angle RNO$ .

The degree measures of three angles of a triangle total $180 ^{\circ }$ , so:

$m \angle NRO + m \angle RNO + m \angle NOR = 180^{\circ }$

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

$2 m \angle RNO + 48^{\circ } = 180^{\circ }$

$2 m \angle RNO = 132^{\circ }$

$m \angle RNO = 66^{\circ }$

Since

$m \angle MNR + m \angle RNO = m \angle MNO$

we have

$m \angle MNR + 66 ^{\circ } = 108^{\circ }$

$m \angle MNR = 42^{\circ }$

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Question

Pentagon is regular. If diagonal $\overline{BE}$ is drawn, which of the following describes Quadrilateral ?

Answer

The figure described is below.

Each of the angles of the pentagon has measure $\frac{180^{\circ } (5-2)}{5} = 108^{\circ }$

$\Delta ABE$ is an isosceles triangle, and $m\angle A = 108^{\circ }$ , so

$m \angle ABE = \frac{1}{2} \left (180^{\circ } - 108^{\circ } \right ) = 36^{\circ }$

and

$m \angle CBE = m \angle CBA - m \angle ABE = 108^{\circ } - 36^{\circ } = 72^{\circ }$

Since

$m \angle C =108^{\circ }$ ,

$m \angle C + m\angle CBE = 108^{\circ } + 72^{\circ } = 180^{\circ }$

and by the parallel postulate,

$\overline{CD} || \overline{BE}$

Quadrilateral has exactly one pair of parallel sides, so it is a trapezoid.

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