Acute / Obtuse Triangles - PSAT Math

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Question

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?

Answer

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

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Question

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

Answer

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

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Question

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?

Answer

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.

Add thirty to both sides.

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.

The answer is 108.

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Question

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.

Answer

The answer is 18

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

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Question

In the picture above, $\overline{AB}$ is a straight line segment. Find the value of .

Answer

A straight line segment has 180 degrees. Therefore, the angle that is not labelled must have:

$180-110 = 70^{\circ}$

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:

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Question

$\textup{If the average of two angles of a triangle is equal to 50 degrees, what}$

$\textup{must be the measure of one of the angles in the triangle?}$

Answer

$\textup{Interior angles must add up to 180 degrees. } \frac{a+b}{2}=50$

$a+b = 100.: :\textup{ Therefore, the third angle must = 80 degrees.}$

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Question

An exterior angle of an isosceles triangle measures $110 ^{\circ }$ . What is the least measure of any of the three interior angles of the triangle?

Answer

The triangle has an exterior angle of $110 ^{\circ }$ , so it has an interior angle of $(180-110)^{\circ } = 70^{\circ }$ . By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles; there are two possible scenarios that fit this criterion:

I: One of the other angles also has measure $70^{\circ }$ .

In this case, since the angles' measures must total $180^{\circ }$ the third angle has measure

$180^{\circ } - (70^{\circ } +70 ^{\circ } ) = 180 ^{\circ } - 140 ^{\circ } = 40 ^{\circ }$

In this case, the least measure is $40 ^{\circ }$ .

II: The other two angles are the congruent angles.

In this case, the other two angles have measures totaling

$180^{\circ } - (70^{\circ } ) = 110 ^{\circ }$ .

They have the same measure, so each has measure half this, or

$\frac{1}{2} \times 110 ^{\circ } = 55^{\circ }$ .

In this case, the least measure is $55^{\circ }$

Therefore, insufficient information is given to answer this question.

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Question

An exterior angle of an isosceles triangle measures $44 ^{\circ }$ . What is the greatest measure of any of the three angles of the triangle?

Answer

The triangle has an exterior angle of $44 ^{\circ }$ , so it has an interior angle of $(180-44)^{\circ } = 136^{\circ }$ . This is an obtuse angle; the other two angles must be acute, and therefore, they will have measure less than $90 ^{\circ }$ - and, subsequently, the $136^{\circ }$ angle will be the one of greatest measure.

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Question

You are given triangles $\Delta MNO$ and $\Delta PQR$ , with and . Which of these statements, along with what you are given, is not enough to prove that $\Delta MNO \cong \Delta PQR$ ?

I) $\Delta MNO$ and $\Delta PQR$ have the same perimeter

II) $m \angle M + m \angle O = m \angle P + m \angle R$

III) $m \angle M + m \angle N = m \angle P + m \angle Q$

Answer

If $\Delta MNO$ and $\Delta PQR$ have the same perimeter, , and , it follows that . The three triangles have the same sidelengths, setting the conditions for the Side-Side-Side Congruence Postulate.

If $m \angle M + m \angle O = m \angle P + m \angle R$ , then, since the sum of the degree measures of both triangles is the same (180 degrees), it follows that $m \angle N = m \angle Q$ . Since $\angle N$ and $\angle Q$ are congruent included angles of congruent sides, this sets the conditions for the SAS Congruence Postulate.

In both of the above cases, it follows that $\Delta MNO \cong \Delta PQR$ .

However, similarly to the previous situation, if $m \angle M + m \angle N = m \angle P + m \angle Q$ , then it follows that $m \angle O = m \ m \angle R$ , meaning that we have congruent sides and congruent nonincluded angles. However, this is not sufficient to prove congruence.

"Statement III" is the correct response.

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Question

$\text{Triangle A}$ and $\text{Triangle B}$ are similar triangles. The perimeter of Triangle A is 45” and the length of two of its sides are 15” and 10”. If the perimeter of Triangle B is 135” and what are lengths of two of its sides?

Answer

The perimeter is equal to the sum of the three sides. In similar triangles, each side is in proportion to its correlating side. The perimeters are also in equal proportion.

Perimeter A = 45” and perimeter B = 135”

The proportion of Perimeter A to Perimeter B is $\frac{45}{135}=\frac{1}{3}$ .

This applies to the sides of the triangle. Therefore to get the any side of Triangle B, just multiply the correlating side by 3.

15” x 3 = 45”

10” x 3 = 30“

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Question

If triangle ABC has vertices (0, 0), (6, 0), and (2, 3) in the xy-plane, what is the area of ABC?

Answer

Sketching ABC in the xy-plane, as pictured here, we see that it has base 6 and height 3. Since the formula for the area of a triangle is 1/2 * base * height, the area of ABC is 1/2 * 6 * 3 = 9.

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Question

The height, , of triangle in the figure is one-fourth the length of . In terms of h, what is the area of triangle ?

Answer

If $\dpi{100} \small h=\frac{1}{4}$ * $\dpi{100} \small \overline{PQ}$ , then the length of $\dpi{100} \small \overline{PQ}$ must be $\dpi{100} \small 4h$ .

Using the formula for the area of a triangle ( $\frac{1}{2}bh$ ), with $\dpi{100} \small b=4h$ , the area of the triangle must be $2h^{2}$ .

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Question

Find the height of a triangle if the area of the triangle = 18 and the base = 4.

Answer

The area of a triangle = (1/2)bh where b is base and h is height. 18 = (1/2)4h which gives us 36 = 4h so h =9.

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Question

A triangle has sides of length 8, 13, and L. Which of the following cannot equal L?

Answer

The sum of the lengths of two sides of a triangle cannot be less than the length of the third side. 8 + 4 = 12, which is less than 13.

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Question

Two sides of a triangle are 20 and 32. Which of the following CANNOT be the third side of this triangle.

Answer

Please remember the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be greater than the third side. Therefore, the correct answer is 10 because the sum of 10 and 20 would not be greater than the third side 32.

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Question

A triangle has sides of length 5, 7, and x. Which of the following can NOT be a value of x?

Answer

The sum of the lengths of any two sides of a triangle must exceed the length of the third side; therefore, 5+7 > x, which cannot happen if x = 13.

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Question

Two sides of a triangle have lengths 4 and 7. Which of the following represents the set of all possible lengths of the third side, x?

Answer

The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.

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Question

If two sides of a triangle have lengths 8 and 10, what could the length of the third side NOT be?

Answer

According to the Triangle Inequality Theorem, the sums of the lengths of any two sides of a triangle must be greater than the length of the third side. Since 10 + 8 is 18, the only length out of the answer choices that is not possible is 19.

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Question

The lengths of two sides of a triangle are 9 and 7. Which of the following could be the length of the third side?

Answer

Let us call the third side x. According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be larger than the other two sides. Thus, all of the following must be true:

x + 7 > 9

x + 9 > 7

7 + 9 > x

We can solve these three inequalities to determine the possible values of x.

x + 7 > 9

Subtract 7 from both sides.

x > 2

Now, we can look at x + 9 > 7. Subtracting 9 from both sides, we obtain

x > –2

Finally, 7 + 9 > x, which means that 16 > x.

Therefore, x must be greater than 2, greater than –2, but also less than 16. The only number that satisfies all of these requirements is 12.

The answer is 12.

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Question

The lengths of a triangle are 8, 12, and x. Which of the following inequalities shows all of the possible values of x?

Answer

According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be greater (not greater than or equal) than the remaining side. Thus, the following inequalities must all be true:

x + 8 > 12

x + 12 > 8

8 + 12 > x

Let's solve each inequality.

x + 8 > 12

Subtract 8 from both sides.

x > 4

Next, let's look at the inequality x + 12 > 8

x + 12 > 8

Subtract 12 from both sides.

x > –4

Lastly, 8 + 12 > x, which means that x < 20.

This means that x must be less than twenty, but greater than 4 and greater than –4. Since any number greater than 4 is also greater than –4, we can exclude the inequality x > –4.

To summarize, x must be greater than 4 and less than 20. We can write this as 4 < x < 20.

The answer is 4 < x < 20.

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