How to find an angle in an acute / obtuse triangle - PSAT Math
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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
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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.
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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?
Tap to reveal 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.
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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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
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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.
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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
Tap to reveal answer
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
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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?
Tap to reveal 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
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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Which of the following can NOT be the angles of a triangle?
Which of the following can NOT be the angles of a triangle?
Tap to reveal answer
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.
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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?
Tap to reveal 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.
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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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.
Tap to reveal 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
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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In the picture above, is a straight line segment. Find the value of .
In the picture above,
Tap to reveal answer
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
Tap to reveal answer
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.
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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
Tap to reveal answer
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
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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?
Tap to reveal 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
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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Which of the following can NOT be the angles of a triangle?
Which of the following can NOT be the angles of a triangle?
Tap to reveal answer
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.
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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?
Tap to reveal 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.
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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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.
Tap to reveal 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
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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