### All SAT Math Resources

## Example Questions

### Example Question #131 : Geometry

If the average of the measures of two angles in a triangle is 75^{o}, what is the measure of the third angle in this triangle?

**Possible Answers:**

75°

40°

30°

50°

65°

**Correct answer:**

30°

The sum of the angles in a triangle is 180^{o}: a + b + c = 180

In this case, the average of a and b is 75:

(a + b)/2 = 75, then multiply both sides by 2

(a + b) = 150, then substitute into first equation

150 + c = 180

c = 30

### Example Question #2 : How To Find An Angle In An Acute / Obtuse Triangle

Which of the following can NOT be the angles of a triangle?

**Possible Answers:**

45, 45, 90

30, 60, 90

1, 2, 177

45, 90, 100

30.5, 40.1, 109.4

**Correct answer:**

45, 90, 100

In a triangle, there can only be one obtuse angle. Additionally, all the angle measures must add up to 180.

### Example Question #3 : How To Find An Angle In An Acute / Obtuse Triangle

Let the measures, in degrees, of the three angles of a triangle be x, y, and z. If y = 2z, and z = 0.5x - 30, then what is the measure, in degrees, of the largest angle in the triangle?

**Possible Answers:**

**Correct answer:**108

The measures of the three angles are x, y, and z. Because the sum of the measures of the angles in any triangle must be 180 degrees, we know that x + y + z = 180. We can use this equation, along with the other two equations given, to form this system of equations:

x + y + z = 180

y = 2z

z = 0.5x - 30

If we can solve for both y and x in terms of z, then we can substitute these values into the first equation and create an equation with only one variable.

Because we are told already that y = 2z, we alreay have the value of y in terms of z.

We must solve the equation z = 0.5x - 30 for x in terms of z.

Add thirty to both sides.

z + 30 = 0.5x

Mutliply both sides by 2

2(z + 30) = 2z + 60 = x

x = 2z + 60

Now we have the values of x and y in terms of z. Let's substitute these values for x and y into the equation x + y + z = 180.

(2z + 60) + 2z + z = 180

5z + 60 = 180

5z = 120

z = 24

Because y = 2z, we know that y = 2(24) = 48. We also determined earlier that x = 2z + 60, so x = 2(24) + 60 = 108.

Thus, the measures of the three angles of the triangle are 24, 48, and 108. The question asks for the largest of these measures, which is 108.

The answer is 108.

### Example Question #1 : Acute / Obtuse Triangles

Angles x, y, and z make up the interior angles of a scalene triangle. Angle x is three times the size of y and 1/2 the size of z. How big is angle y.

**Possible Answers:**

108

42

18

54

36

**Correct answer:**

18

The answer is 18

We know that the sum of all the angles is 180. Using the rest of the information given we can write the other two equations:

x + y + z = 180

x = 3y

2x = z

We can solve for y and z in the second and third equations and then plug into the first to solve.

x + (1/3)x + 2x = 180

3[x + (1/3)x + 2x = 180]

3x + x + 6x = 540

10x = 540

x = 54

y = 18

z = 108

### Example Question #1 : Acute / Obtuse Triangles

In the picture above, is a straight line segment. Find the value of .

**Possible Answers:**

**Correct answer:**

A straight line segment has 180 degrees. Therefore, the angle that is not labelled must have:

We know that the sum of the angles in a triangle is 180 degrees. As a result, we can set up the following algebraic equation:

Subtract 70 from both sides:

Divide by 2:

### Example Question #6 : How To Find An Angle In An Acute / Obtuse Triangle

If , , and are measures of three angles of a triangle, what is the value of ?

**Possible Answers:**

**Correct answer:**

Since the sum of the angles of a triangle is , we know that

.

So

and .

### Example Question #7 : How To Find An Angle In An Acute / Obtuse Triangle

Solve each problem and decide which is the best of the choices given.

Solve for .

**Possible Answers:**

**Correct answer:**

To solve for , you must first solve for .

All triangles' angles add up to .

So subtract from to get , the value of .

Angles and are supplementary, meaning they add up to .

Subtract from to get .

, so .

### Example Question #8 : How To Find An Angle In An Acute / Obtuse Triangle

Refer to the above figure. Evaluate .

**Possible Answers:**

**Correct answer:**

is marked with three congruent sides, making it an equilateral triangle, so . This is an exterior angle of , making its measure the sum of those of its remote interior angles; that is,

has congruent sides and , so, by the Isosceles Triangle Theorem, . Substituting for and for :

and form a linear pair and are therefore supplementary - that is, their degree measures total . Setting up the equation

and substituting:

### Example Question #9 : How To Find An Angle In An Acute / Obtuse Triangle

Figure is not drawn to scale.

Refer to the provided figure. Evaluate .

**Possible Answers:**

**Correct answer:**

is an equilateral, so all of its angles - in particular, - measure . This angle is an exterior angle to , and its measure is equal to the sum of those of its two remote interior angles, and , so

Setting and , solve for :

### Example Question #1 : Acute / Obtuse Triangles

If a = 7 and b = 4, which of the following could be the perimeter of the triangle?

I. 11

II. 15

III. 25

**Possible Answers:**

II and III Only

II Only

I and II Only

I Only

I, II and III

**Correct answer:**

II Only

Consider the perimeter of a triangle:

P = a + b + c

Since we know a and b, we can find c.

In I:

11 = 7 + 4 + c

11 = 11 + c

c = 0

Note that if c = 0, the shape is no longer a trial. Thus, we can eliminate I.

In II:

15 = 7 + 4 + c

15 = 11 + c

c = 4.

This is plausible given that the other sides are 7 and 4.

In III:

25 = 7 + 4 + c

25 = 11 + c

c = 14.

It is not possible for one side of a triangle to be greater than the sum of both of the other sides, so eliminate III.

Thus we are left with only II.