### All PSAT Math Resources

## Example Questions

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

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

**Possible Answers:**

8

6

9

5

7

**Correct answer:**

9

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 : Other Polygons

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:**

128.57 degrees

257.14 degrees

180 degrees

252 degrees

154.28 degrees

**Correct answer:**

128.57 degrees

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 #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?

**Possible Answers:**

A/D < B/C

AD = BC

None of the answers.

A * C > B * D

**Correct answer:**

None of the answers.

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*?

**Possible Answers:**

50 degrees

There is more than one correct answer

80 degrees

65 degrees

95 degrees

**Correct answer:**

95 degrees

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 *B* = *x* degrees, and the measure of angle *C* = *y* degrees. What is the value of *y* in terms of *x*?

**Possible Answers:**

70 + *x*

*x* – 70

70 – *x*

110 + *x*

110 – *x*

**Correct answer:**

110 – *x*

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:**

180

135

175

150

120

**Correct answer:**

150

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 #1 : Other Polygons

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

**Possible Answers:**

30

40

60

45

50

**Correct answer:**

45

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 #1 : How To Find An Angle In A Polygon

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

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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

Note: Figure NOT drawn to scale.

Refer to the above figure. is equilateral and Pentagon is regular.

Evaluate .

**Possible Answers:**

**Correct answer:**

By angle addition,

is an angle of a reguar pentagon, so its measure is .

To find , first we find .

By angle addition,

is an angle of a regular pentagon and has measure .

, as an angle of an equilateral triangle, has measure .

is equilateral, so ; Pentagon is regular, so . Therefore, , and by the Isosceles Triangle Theorem, .

The degree measures of three angles of a triangle total , so:

Since

we have

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