### All PSAT Math Resources

## Example Questions

### Example Question #91 : Right Triangles

Acute angles x and y are inside a right triangle. If x is four less than one third of 21, what is y?

**Possible Answers:**

87

3

7

18

90

**Correct answer:**

87

We know that the sum of all the angles must be 180 and we already know one angle is 90, leaving the sum of x and y to be 90.

Solve for x to find y.

One third of 21 is 7. Four less than 7 is 3. So if angle x is 3 then that leaves 87 for angle y.

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

If a right triangle has one leg with a length of 4 and a hypotenuse with a length of 8, what is the measure of the angle between the hypotenuse and its other leg?

**Possible Answers:**

90

60

45

65

30

**Correct answer:**

30

The first thing to notice is that this is a 30^{o}:60^{o}:90^{o} triangle. If you draw a diagram, it is easier to see that the angle that is asked for corresponds to the side with a length of 4. This will be the smallest angle. The correct answer is 30.

### Example Question #2 : Right Triangles

In the figure above, what is the positive difference, in degrees, between the measures of angle *ACB* and angle *CBD*?

**Possible Answers:**

30

20

10

40

50

**Correct answer:**

10

In the figure above, angle *ADB* is a right angle. Because side *AC* is a straight line, angle *CDB* must also be a right angle.

Let’s examine triangle *ADB*. The sum of the measures of the three angles must be 180 degrees, and we know that angle *ADB* must be 90 degrees, since it is a right angle. We can now set up the following equation.

*x* + *y* + 90 = 180

Subtract 90 from both sides.

*x* + *y* = 90

Next, we will look at triangle *CDB*. We know that angle *CDB* is also 90 degrees, so we will write the following equation:

*y* – 10 + 2*x* – 20 + 90 = 180

*y* + 2*x* + 60 = 180

Subtract 60 from both sides.

*y* + 2*x* = 120

We have a system of equations consisting of *x* + *y* = 90 and *y* + 2*x* = 120. We can solve this system by solving one equation in terms of *x* and then substituting this value into the second equation. Let’s solve for *y* in the equation *x* + *y* = 90.

*x* + *y* = 90

Subtract *x* from both sides.

*y* = 90 – *x*

Next, we can substitute 90 – *x* into the equation *y* + 2*x* = 120.

(90 – *x*) + 2*x* = 120

90 + *x* = 120

*x* = 120 – 90 = 30

*x* = 30

Since *y* = 90 – *x*, *y* = 90 – 30 = 60.

The question ultimately asks us to find the positive difference between the measures of *ACB* and *CBD*. The measure of *ACB* = 2*x* – 20 = 2(30) – 20 = 40 degrees. The measure of *CBD* = *y* – 10 = 60 – 10 = 50 degrees. The positive difference between 50 degrees and 40 degrees is 10.

The answer is 10.

### Example Question #2 : Triangles

Which of the following sets of line-segment lengths can form a triangle?

**Possible Answers:**

**Correct answer:**

In any given triangle, the sum of any two sides is greater than the third. The incorrect answers have the sum of two sides equal to the third.

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

In right , and .

What is the value of ?

**Possible Answers:**

48

30

32

24

36

**Correct answer:**

36

There are 180 degrees in every triangle. Since this triangle is a right triangle, one of the angles measures 90 degrees.

Therefore, .

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

Refer to the above diagram. Which of the following gives a valid alternative name for ?

**Possible Answers:**

All four of the other choices gives a valid alternative name for the triangle.

**Correct answer:**

All four of the other choices gives a valid alternative name for the triangle.

A triangle can be named after its three vertices in any order, so all of the choices given are valid.

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