# SAT Math : How to find an angle in a polygon

## Example Questions

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### Example Question #1 : 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?

A/D < B/C

A * C > B * D

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 #1 : How To Find An Angle In A Polygon

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

65 degrees

80 degrees

There is more than one correct answer

50 degrees

95  degrees

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 #1 : How To Find An Angle In A Polygon

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?

70 – x

110 + x

110 – x

x – 70

70 + x

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?

120

150

135

175

180

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.

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

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

60

45

40

30

50

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.

### Example Question #9 : Other Polygons

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

Explanation:

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

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

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

Explanation:

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

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

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

Explanation:

The 5 angles of a pentagon add to 540

The 6 angles of a hexagon add to 720

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

Find the sum of the interior angles in a nonagon.

Explanation:

To solve, simply use the formula for the total degrees in a polygon, where n is the number of vertices.

In this particular case, a nonagon is a shape with nine sides and thus nine vertices.

Thus,

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

Which of the following cannot be the measure of an exterior angle of a regular polygon?

Explanation:

If one exterior angle is taken at each vertex of any convex polygon, the sum of their measures is . In a regular polygon - one with congruent sides and congruent interior angles, each exterior angle is congruent to one another. If the polygon has  sides, each exterior angle has measure .

Given the common measure

Multiplying both sides by :

and

Since  is equal to a number of sides, it is a whole number. Thus, we are looking for a value of  which, when we divide 360 by it, yields a non-whole result. We see that  is the correct choice, since'

A quick check confirms that 360 divided by 8, 10, 12, or 15 yields a whole result.

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