### All PSAT Math Resources

## Example Questions

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

You are given triangles and , with and . Which of these statements, along with what you are given, is* not *enough to prove that ?

I) and have the same perimeter

II)

III)

**Possible Answers:**

Statement I only

Statement III only

None of these statements is enough to prove congruence.

Statement II only

Any of the three statements is enough to prove congruence.

**Correct answer:**

Statement III only

If and have the same perimeter, , and , it follows that . The three triangles have the same sidelengths, setting the conditions for the Side-Side-Side Congruence Postulate.

If , then, since the sum of the degree measures of both triangles is the same (180 degrees), it follows that . Since and are congruent included angles of congruent sides, this sets the conditions for the SAS Congruence Postulate.

In both of the above cases, it follows that .

However, similarly to the previous situation, if , then it follows that , meaning that we have congruent sides and congruent *nonincluded *angles. However, this is not sufficient to prove congruence.

"Statement III" is the correct response.

### 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:**

I Only

II and III Only

II Only

I, II and III

I and II Only

**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.

### Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

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:**

40°

75°

30°

65°

50°

**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 #1 : How To Find An Angle In An Acute / Obtuse Triangle

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

**Possible Answers:**

30.5, 40.1, 109.4

30, 60, 90

1, 2, 177

45, 45, 90

45, 90, 100

**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 #321 : Plane Geometry

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 : How To Find An Angle In An Acute / Obtuse Triangle

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:**

42

36

108

18

54

**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 #1 : How To Find An Angle In An Acute / Obtuse Triangle

An exterior angle of an isosceles triangle measures . What is the least measure of any of the three interior angles of the triangle?

**Possible Answers:**

Insufficient information is given to answer this question.

**Correct answer:**

Insufficient information is given to answer this question.

The triangle has an exterior angle of , so it has an interior angle of . By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles; there are two possible scenarios that fit this criterion:

I: One of the other angles also has measure .

In this case, since the angles' measures must total the third angle has measure

In this case, the least measure is .

II: The other two angles are the congruent angles.

In this case, the other two angles have measures totaling

.

They have the same measure, so each has measure half this, or

.

In this case, the least measure is

Therefore, insufficient information is given to answer this question.

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

An exterior angle of an isosceles triangle measures . What is the greatest measure of any of the three angles of the triangle?

**Possible Answers:**

Insufficient information is given to answer this question.

**Correct answer:**

The triangle has an exterior angle of , so it has an interior angle of . This is an obtuse angle; the other two angles must be acute, and therefore, they will have measure less than - and, subsequently, the angle will be the one of greatest measure.

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

**Possible Answers:**

**Correct answer:**