Acute / Obtuse Triangles - PSAT Math
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A triangle has sides of length 5, 7, and x. Which of the following can NOT be a value of x?
A triangle has sides of length 5, 7, and x. Which of the following can NOT be a value of x?
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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.
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.
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?
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?
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The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.
The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.
If two sides of a triangle have lengths 8 and 10, what could the length of the third side NOT be?
If two sides of a triangle have lengths 8 and 10, what could the length of the third side NOT be?
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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.
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.
The lengths of two sides of a triangle are 9 and 7. Which of the following could be the length of the third side?
The lengths of two sides of a triangle are 9 and 7. Which of the following could be the length of the third side?
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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.
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.
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?
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?
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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
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
Which of the following can NOT be the angles of a triangle?
Which of the following can NOT be the angles of a triangle?
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In a triangle, there can only be one obtuse angle. Additionally, all the angle measures must add up to 180.
In a triangle, there can only be one obtuse angle. Additionally, all the angle measures must add up to 180.
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?
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?
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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.
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.
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.
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.
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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
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
In the picture above, 
is a straight line segment. Find the value of 
.
In the picture above,
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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:
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:
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An exterior angle of an isosceles triangle measures 
. What is the least measure of any of the three interior angles of the triangle?
An exterior angle of an isosceles triangle measures
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The triangle has an exterior angle of 
, so it has an interior angle of 
. 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 
.
In this case, since the angles' measures must total 
the third angle has measure
In this case, the least measure is 
.
II: The other two angles are the congruent angles.
In this case, the other two angles have measures totaling
.
They have the same measure, so each has measure half this, or
.
In this case, the least measure is 
Therefore, insufficient information is given to answer this question.
The triangle has an exterior angle of
I: One of the other angles also has measure
In this case, since the angles' measures must total
In this case, the least measure is
II: The other two angles are the congruent angles.
In this case, the other two angles have measures totaling
They have the same measure, so each has measure half this, or
In this case, the least measure is
Therefore, insufficient information is given to answer this question.
An exterior angle of an isosceles triangle measures 
. What is the greatest measure of any of the three angles of the triangle?
An exterior angle of an isosceles triangle measures
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The triangle has an exterior angle of 
, so it has an interior angle of 
. This is an obtuse angle; the other two angles must be acute, and therefore, they will have measure less than 
- and, subsequently, the 
angle will be the one of greatest measure.
The triangle has an exterior angle of
You are given triangles 
and 
, with 
and 
. Which of these statements, along with what you are given, is not enough to prove that 
?
I) 
and 
have the same perimeter
II) 
III) 
You are given triangles
I)
II)
III)
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If 
and 
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 
, then, since the sum of the degree measures of both triangles is the same (180 degrees), it follows that 
. Since 
and 
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 
.
However, similarly to the previous situation, if 
, then it follows that 
, meaning that we have congruent sides and congruent nonincluded angles. However, this is not sufficient to prove congruence.
"Statement III" is the correct response.
If
If
In both of the above cases, it follows that
However, similarly to the previous situation, if
"Statement III" is the correct response.
and 
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?
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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 
.
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“
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
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“
If triangle ABC has vertices (0, 0), (6, 0), and (2, 3) in the xy-plane, what is the area of ABC?
If triangle ABC has vertices (0, 0), (6, 0), and (2, 3) in the xy-plane, what is the area of ABC?
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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.
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.
The height, 
, of triangle 
in the figure is one-fourth the length of 
. In terms of h, what is the area of triangle 
?
The height, 
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If h=\frac{1}{4}$ *$\overline{PQ}$, then the length of $\overline{PQ}$ must be 4h.
Using the formula for the area of a triangle ($\frac{1}{2}$bh), with b=4h, the area of the triangle must be $2h^{2}$.
If h=\frac{1}{4}$ *$\overline{PQ}$, then the length of $\overline{PQ}$ must be 4h.
Using the formula for the area of a triangle ($\frac{1}{2}$bh), with b=4h, the area of the triangle must be $2h^{2}$.
Find the height of a triangle if the area of the triangle = 18 and the base = 4.
Find the height of a triangle if the area of the triangle = 18 and the base = 4.
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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.
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.
A triangle has sides of length 8, 13, and L. Which of the following cannot equal L?
A triangle has sides of length 8, 13, and L. Which of the following cannot equal L?
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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.
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.
Two sides of a triangle are 20 and 32. Which of the following CANNOT be the third side of this triangle.
Two sides of a triangle are 20 and 32. Which of the following CANNOT be the third side of this triangle.
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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.
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.
The lengths of a triangle are 8, 12, and x. Which of the following inequalities shows all of the possible values of x?
The lengths of a triangle are 8, 12, and x. Which of the following inequalities shows all of the possible values of x?
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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.
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.