# SAT Math : Acute / Obtuse Triangles

## Example Questions

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### 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 75o, what is the measure of the third angle in this triangle?

75°

65°

50°

40°

30°

30°

Explanation:

The sum of the angles in a triangle is 180o:  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 #2 : How To Find An Angle In An Acute / Obtuse Triangle

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

45, 45, 90

1, 2, 177

30, 60, 90

45, 90, 100

30.5, 40.1, 109.4

45, 90, 100

Explanation:

In a triangle, there can only be one obtuse angle. Additionally, all the angle measures must add up to 180.

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

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?

48
96
60
108
30
Explanation:

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.

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.

### Example Question #4 : 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.

18

108

54

36

42

18

Explanation:

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

In the picture above,  is a straight line segment. Find the value of .

Explanation:

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 #102 : Triangles

If , and  are measures of three angles of a triangle, what is the value of

Explanation:

Since the sum of the angles of a triangle is , we know that

.

So

and .

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

Solve each problem and decide which is the best of the choices given.

Solve for .

Explanation:

To solve for , you must first solve for .

All triangles' angles add up to .

So subtract  from  to get , the value of .

Angles  and  are supplementary, meaning they add up to .

Subtract  from  to get .

, so .

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

Refer to the above figure. Evaluate .

Explanation:

is marked with three congruent sides, making it an equilateral triangle, so . This is an exterior angle of , making its measure the sum of those of its remote interior angles; that is,

has congruent sides  and , so, by the Isosceles Triangle Theorem, . Substituting   for  and  for :

and  form a linear pair and are therefore supplementary - that is, their degree measures total . Setting up the equation

and substituting:

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

Figure is not drawn to scale.

Refer to the provided figure. Evaluate .

Explanation:

is an equilateral, so all of its angles - in particular,   - measure . This angle is an exterior angle to ,  and its measure is equal to the sum of those of its two remote interior angles,  and , so

Setting  and , solve for :

### Example Question #1 : How To Find The Perimeter Of An Acute / Obtuse Triangle

If a = 7 and b = 4, which of the following could be the perimeter of the triangle?

I. 11

II. 15

III. 25

II and III Only

II Only

I Only

I and II Only

I, II and III

II Only

Explanation:

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.

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