SAT Math : How to find an angle in a polygon

Study concepts, example questions & explanations for SAT Math

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Example Questions

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Example Question #371 : Plane Geometry

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

 

Possible Answers:

7

9

8

6

5

Correct answer:

9

Explanation:

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

Example Question #1 : How To Find An Angle In A Polygon

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

Possible Answers:

257.14 degrees

252 degrees

180 degrees

154.28 degrees

128.57 degrees

Correct answer:

128.57 degrees

Explanation:

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

Example Question #2 : How To Find An Angle In A Polygon

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

Possible Answers:

None of the answers.

A * C > B * D

A/D < B/C

AD = BC

Correct answer:

None of the answers.

Explanation:

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.

Example Question #374 : Plane Geometry

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

Possible Answers:

95  degrees

80 degrees

There is more than one correct answer

50 degrees

65 degrees

Correct answer:

95  degrees

Explanation:

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.

Example Question #375 : Plane Geometry

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

Possible Answers:

70 – x

110 – x

110 + x

70 + x

x – 70

Correct answer:

110 – x

Explanation:

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.

Example Question #1 : How To Find An Angle In A Polygon

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

Possible Answers:

135

120

150

180

175

Correct answer:

150

Explanation:

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.

Example Question #5 : Other Polygons

Octagon

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

Possible Answers:

40

50

30

60

45

Correct answer:

45

Explanation:

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. 

Example Question #2 : How To Find An Angle In A Polygon

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

Possible Answers:

90

180

75

135

150

Correct answer:

135

Explanation:

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

Example Question #3 : How To Find An Angle In A Polygon

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

Possible Answers:

120

90

35

60

45

Correct answer:

45

Explanation:

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

Example Question #373 : Plane Geometry

What is the average (arithmetic mean) of all 15 interior angles of a quadrilateral, pentagon, and hexagon?

Possible Answers:

Correct answer:

Explanation:

The 4 angles of a quadrilateral add to 360

The 5 angles of a pentagon add to 540

The 6 angles of a hexagon add to 720

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