# ACT Math : Triangles

## Example Questions

### Example Question #7 : Acute / Obtuse Isosceles Triangles

What is the perimeter of an isosceles triangle with a vertex of  degrees and two sides equal to

Explanation:

Based on the description of your triangle, you can draw the following figure:

You can do this because you know:

1. The two equivalent sides are given.
2. Since a triangle is  degrees, you have only  or  degrees left for the two angles of equal size. Therefore, those two angles must be  degrees and  degrees.

Now, based on the properties of an isosceles triangle, you can draw the following as well:

Based on your standard reference  triangle, you know:

Therefore,  is .

This means that  is  and the total base of the triangle is .

Therefore, the perimeter of the triangle is:

### Example Question #8 : Acute / Obtuse Isosceles Triangles

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , and . Two of the angles in Triangle A each measure . Triangle B's sides measure , and . What is the measure of the smallest angle in Triangle B?

Explanation:

Because the interior angles of a triangle add up to , and two of Triangle A's interior angles measure , we must simply add the two given angles and subtract from  to find the missing angle:

Therefore, the missing angle (and the smallest) from Triangle A measures . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also .

The side measurements presented in the question are not needed to find the answer!

### Example Question #9 : Acute / Obtuse Isosceles Triangles

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of  and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Explanation:

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side (). Because the height line cuts the base in half, however, we must use  for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of  and the height of , we use the Pythagorean Theorem to solve for :

Therefore, the two congruent sides of Triangle A measure ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is  and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of  to the side we're trying to find (), as follows:

Therefore, the length of the congruent sides of Triangle B is .

### Example Question #10 : Acute / Obtuse Isosceles Triangles

Isosceles triangles  and  share common side  is an obtuse triangle with sides  is also an obtuse isosceles triangle, where . What is the measure of ?

Explanation:

In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side  is not one of the two congruent sides of , so we cannot state congruence with .

### 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?

20°

30°

Cannot be determined from the given information

15°

25°

20°

Explanation:

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?

40 degrees

100 degrees

140 degrees

70 degrees

70 degrees

Explanation:

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?

Explanation:

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?

Explanation:

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 ?

Explanation:

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?

Explanation:

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.