### All ACT Math Resources

## Example Questions

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

Points A and B lie on a circle centered at Z, where central angle <AZB measures 140°. What is the measure of angle <ZAB?

**Possible Answers:**

Cannot be determined from the given information

20°

15°

30°

25°

**Correct answer:**

20°

Because line segments ZA and ZB are radii of the circle, they must have the same length. That makes triangle ABZ an isosceles triangle, with <ZAB and <ZBA having the same measure. Because the three angles of a triangle must sum to 180°, you can express this in the equation:

140 + 2x = 180 **-->** 2x = 40 **-->** x = 20

### Example Question #11 : Acute / Obtuse Isosceles Triangles

Triangle FGH has equal lengths for FG and GH; what is the measure of ∠F, if ∠G measures 40 degrees?

**Possible Answers:**

140 degrees

40 degrees

None of the other answers

70 degrees

100 degrees

**Correct answer:**

70 degrees

It's good to draw a diagram for this; we know that it's an isosceles triangle; remember that the angles of a triangle total 180 degrees.

Angle G for this triangle is the one angle that doesn't correspond to an equal side of the isosceles triangle (opposite side to the angle), so that means ∠F = ∠H, and that ∠F + ∠H + 40 = 180,

By substitution we find that ∠F * 2 = 140 and angle F = 70 degrees.

### Example Question #6 : Triangles

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

**Possible Answers:**

**Correct answer:**

An isosceles triangle has two congruent base angles and one vertex angle. Each triangle contains . Let = base angle, so the equation becomes . Solving for gives

### Example Question #7 : Triangles

In an isosceles triangle the base angle is five less than twice the vertex angle. What is the sum of the vertex angle and 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 = base angle

So the equation to solve becomes

or

Thus the vertex angle is 38 and the base angle is 71 and their sum is 109.

### Example Question #161 : Triangles

Sides and in this triangle are equal. What is the measure of ?

**Possible Answers:**

**Correct answer:**

This triangle has an angle of . We also know it has another angle of at because the two sides are equal. Adding those two angles together gives us total. Since a triangle has total, we subtract 130 from 180 and get 50.

### Example Question #11 : Acute / Obtuse Isosceles Triangles

An isosceles triangle has a base angle that is six more than three times 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 = vertex angle and = base angle.

Then the equation to solve becomes

or

.

Solving for gives a vertex angle of 24 degrees and a base angle of 78 degrees.

### Example Question #31 : Isosceles Triangles

The base angle of an isosceles triangle is thirteen more than three times the vertex angle. What is the difference between the vertex angle and the base angle?

**Possible Answers:**

**Correct answer:**

Every triangle has . An isosceles triangle has one vertex ange, and two congruent base angles.

Let be the vertex angle and be the base angle.

The equation to solve becomes , since the base angle occurs twice.

Now we can solve for the vertex angle.

The difference between the vertex angle and the base angle is .

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

A particular acute isosceles triangle has an internal angle measuring . Which of the following must be the other two angles?

**Possible Answers:**

**Correct answer:**

By definition, an acute isosceles triangle will have at least two sides (and at least two corresponding angles) that are congruent, and no angle will be greater than . Addtionally, like all triangles, the three angles will sum to . Thus, of our two answers which sum to , only is valid, as would violate the "acute" part of the definition.

### Example Question #2021 : Hspt Mathematics

In triangle ABC, Angle A = x degrees, Angle B = 2x degrees, and Angle C = 3x+30 degrees. How many degrees is Angle B?

**Possible Answers:**

45°

105°

30°

25°

50°

**Correct answer:**

50°

Because the interior angles of a triangle add up to 180°, we can create an equation using the variables given in the problem: x+2x+(3x+30)=180. This simplifies to 6X+30=180. When we subtract 30 from both sides, we get 6x=150. Then, when we divide both sides by 6, we get x=25. Because Angle B=2x degrees, we multiply 25 times 2. Thus, Angle B is equal to 50°. If you got an answer of 25, you may have forgotten to multiply by 2. If you got 105, you may have found Angle C instead of Angle B.

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