### All PSAT Math Resources

## Example Questions

### Example Question #162 : Plane Geometry

Triangle ABC has angle measures as follows:

What is ?

**Possible Answers:**

44

19

57

90

79

**Correct answer:**

57

The sum of the measures of the angles of a triangle is 180.

Thus we set up the equation

After combining like terms and cancelling, we have

Thus

### Example Question #163 : Plane Geometry

The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?

**Possible Answers:**

**Correct answer:**

Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = the vertex angle and = the base angle

So the equation to solve becomes

Thus the vertex angle is 34 and the base angles are 73.

### Example Question #164 : Plane Geometry

The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?

**Possible Answers:**

**Correct answer:**

Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle

So the equation to solve becomes .

### Example Question #1 : Triangles

The base angle of an isosceles triangle is ten less than twice the vertex angle. What is the vertex angle?

**Possible Answers:**

**Correct answer:**

Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle

So the equation to solve becomes

So the vertex angle is 40 and the base angles is 70

### Example Question #166 : Plane Geometry

The base angle of an isosceles triangle is 10 more than twice the vertex angle. What is the vertex angle?

**Possible Answers:**

**Correct answer:**

Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = the vertex angle and = the base angle

So the equation to solve becomes

The vertex angle is 32 degrees and the base angle is 74 degrees

### Example Question #167 : Plane Geometry

In an isosceles triangle, the vertex angle is 15 less than the base angle. What is the base angle?

**Possible Answers:**

**Correct answer:**

Let = base angle and = vertex angle

So the equation to solve becomes

Thus, 65 is the base angle and 50 is the vertex angle.

### Example Question #168 : Plane Geometry

In an isosceles triangle the vertex angle is half the base angle. What is the vertex angle?

**Possible Answers:**

**Correct answer:**

Let = base angle and = vertex angle

So the equation to solve becomes , thus is the base angle and is the vertex angle.

### Example Question #169 : Plane Geometry

If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is , which of the following is the measure of one of the angles of the triangle?

**Possible Answers:**

**Correct answer:**

Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:

Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).

70 + 70 + 40 equals 180 also checks out.

Since 70° is not an answer choice for us, we know that the **40°** must be one of the angles.

### Example Question #2 : Triangles

The base angle of an isosceles triangle is . What is the vertex angle?

**Possible Answers:**

**Correct answer:**

Solve the equation for x to find the measure of the vertex angle.

x = 180 - 27 - 27

x = 126

Therefore the measure of the vertex angle is .

### Example Question #3 : Triangles

Two sides of an isosceles triangle are 20 and 30. What is the difference of the largest and the smallest possible perimeters?

**Possible Answers:**

30

0

The answer cannot be determined

15

10

**Correct answer:**

10

The trick here is that we don't know which is the repeated side. Our possible triangles are therefore 20 + 20 + 30 = 70 or 30 + 30 + 20 = 80. The difference is therefore 80 – 70 or 10.

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