### All ACT Math Resources

## Example Questions

### Example Question #4 : 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:**

20°

30°

Cannot be determined from the given information

15°

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 #5 : 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:**

40 degrees

100 degrees

140 degrees

None of the other answers

70 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 : Acute / Obtuse Isosceles 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 : Acute / Obtuse Isosceles 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 #8 : Acute / Obtuse Isosceles 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 #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

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

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 #4 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

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:**

50°

105°

45°

25°

30°

**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.

### Example Question #11 : Isosceles Triangles

A triangle has a perimeter of inches with one side of length inches. If the remaining two sides have lengths in a ratio of , what is length of the shortest side of the triangle?

**Possible Answers:**

**Correct answer:**

The answer is .

Since we know that the permieter is inches and one side is inches, it can be determined that the remaining two sides must combine to be inches. The ratio of the remaining two sides is which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation , and divide both sides by which means . The ratio of the remaining side lengths then becomes or . We now know the 3 side lengths are .

is the shortest side and thus the answer.

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