Isosceles Triangles - PSAT Math
Card 1 of 105
A triangle with two equal angles is called a(n) .
A triangle with two equal angles is called a(n) .
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An isoceles triangle is a triangle that has at least two congruent sides (and therefore, at least two congruent angles as well).
An isoceles triangle is a triangle that has at least two congruent sides (and therefore, at least two congruent angles as well).
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An isosceles triangle has an area of 12. If the ratio of the base to the height is 3:2, what is the length of the two equal sides?
An isosceles triangle has an area of 12. If the ratio of the base to the height is 3:2, what is the length of the two equal sides?
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Area of a triangle is ½ x base x height. Since base:height = 3:2, base = 1.5 height. Area = 12 = ½ x 1.5 height x height or 24/1.5 = height2. Height = 4. Base = 1.5 height = 6. Half the base and the height form the legs of a right triangle, with an equal leg of the isosceles triangle as the hypotenuse. This is a 3-4-5 right triangle.
Area of a triangle is ½ x base x height. Since base:height = 3:2, base = 1.5 height. Area = 12 = ½ x 1.5 height x height or 24/1.5 = height2. Height = 4. Base = 1.5 height = 6. Half the base and the height form the legs of a right triangle, with an equal leg of the isosceles triangle as the hypotenuse. This is a 3-4-5 right triangle.
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Two sides of a triangle each have length 6. All of the following could be the length of the third side EXCEPT
Two sides of a triangle each have length 6. All of the following could be the length of the third side EXCEPT
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This question is about the Triangle Inequality, which states that in a triangle with two sides A and B, the third side must be greater than the absolute value of the difference between A and B and smaller than the sum of A and B.
Applying the Triangle Inequality to this problem, we see that the third side must be greater than the absolute value of the difference between the other two sides, which is |6-6|=0, and smaller than the sum of the two other sides, which is 6+6=12. The only answer choice that does not satisfy this range of possible values is 12 since the third side must be LESS than 12.
This question is about the Triangle Inequality, which states that in a triangle with two sides A and B, the third side must be greater than the absolute value of the difference between A and B and smaller than the sum of A and B.
Applying the Triangle Inequality to this problem, we see that the third side must be greater than the absolute value of the difference between the other two sides, which is |6-6|=0, and smaller than the sum of the two other sides, which is 6+6=12. The only answer choice that does not satisfy this range of possible values is 12 since the third side must be LESS than 12.
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If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is 
, which of the following is the measure of one of the angles of the triangle?
If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is
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Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:
Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).
70 + 70 + 40 equals 180 also checks out.
Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.
Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:
Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).
70 + 70 + 40 equals 180 also checks out.
Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.
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Triangle ABC has angle measures as follows:
mangle ABC=4x+3
mangle ACB=2x+6
mangle BAC=3x
What is mangle BAC?
Triangle ABC has angle measures as follows:
mangle ABC=4x+3
mangle ACB=2x+6
mangle BAC=3x
What is mangle BAC?
Tap to reveal answer
The sum of the measures of the angles of a triangle is 180.
Thus we set up the equation 4x+3+2x+6+3x=180
After combining like terms and cancelling, we have 9x=171rightarrow x=19
Thus mangle BAC=3x=57
The sum of the measures of the angles of a triangle is 180.
Thus we set up the equation 4x+3+2x+6+3x=180
After combining like terms and cancelling, we have 9x=171rightarrow x=19
Thus mangle BAC=3x=57
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The base angle of an isosceles triangle is $27^{circ}$. What is the vertex angle?
The base angle of an isosceles triangle is $27^{circ}$. What is the vertex angle?
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Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Solve the equation 27+27+x=180 for x to find the measure of the vertex angle.
x = 180 - 27 - 27
x = 126
Therefore the measure of the vertex angle is $126^{circ}$.
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Solve the equation 27+27+x=180 for x to find the measure of the vertex angle.
x = 180 - 27 - 27
x = 126
Therefore the measure of the vertex angle is $126^{circ}$.
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The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?
The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?
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Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let x = the vertex angle and 2x+5 = the base angle
So the equation to solve becomes x+(2x+5)+(2x+5)=180
Thus the vertex angle is 34 and the base angles are 73.
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let x = the vertex angle and 2x+5 = the base angle
So the equation to solve becomes x+(2x+5)+(2x+5)=180
Thus the vertex angle is 34 and the base angles are 73.
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The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?
The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?
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Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let 
= vertex angle and 
= base angle
So the equation to solve becomes 
.
Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let
So the equation to solve becomes
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The base angle of an isosceles triangle is ten less than twice the vertex angle. What is the vertex angle?
The base angle of an isosceles triangle is ten less than twice the vertex angle. What is the vertex angle?
Tap to reveal answer
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let 
= vertex angle and 
= base angle
So the equation to solve becomes 
So the vertex angle is 40 and the base angles is 70
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let
So the equation to solve becomes
So the vertex angle is 40 and the base angles is 70
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The base angle of an isosceles triangle is 10 more than twice the vertex angle. What is the vertex angle?
The base angle of an isosceles triangle is 10 more than twice the vertex angle. What is the vertex angle?
Tap to reveal answer
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let 
= the vertex angle and 
= the base angle
So the equation to solve becomes 
The vertex angle is 32 degrees and the base angle is 74 degrees
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let
So the equation to solve becomes
The vertex angle is 32 degrees and the base angle is 74 degrees
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In an isosceles triangle, the vertex angle is 15 less than the base angle. What is the base angle?
In an isosceles triangle, the vertex angle is 15 less than the base angle. What is the base angle?
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Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let 
= base angle and 
= vertex angle
So the equation to solve becomes 
Thus, 65 is the base angle and 50 is the vertex angle.
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let
So the equation to solve becomes
Thus, 65 is the base angle and 50 is the vertex angle.
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In an isosceles triangle the vertex angle is half the base angle. What is the vertex angle?
In an isosceles triangle the vertex angle is half the base angle. What is the vertex angle?
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Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let x = base angle and 0.5x = vertex angle
So the equation to solve becomes x+x+0.5x=180, thus x=72 is the base angle and 0.5x=36 is the vertex angle.
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let x = base angle and 0.5x = vertex angle
So the equation to solve becomes x+x+0.5x=180, thus x=72 is the base angle and 0.5x=36 is the vertex angle.
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What is the area of a square that has a diagonal whose endpoints in the coordinate plane are located at (-8, 6) and (2, -4)?
What is the area of a square that has a diagonal whose endpoints in the coordinate plane are located at (-8, 6) and (2, -4)?
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A triangle with two equal angles is called a(n) .
A triangle with two equal angles is called a(n) .
Tap to reveal answer
An isoceles triangle is a triangle that has at least two congruent sides (and therefore, at least two congruent angles as well).
An isoceles triangle is a triangle that has at least two congruent sides (and therefore, at least two congruent angles as well).
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If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is , which of the following is the measure of one of the angles of the triangle?
If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is
Tap to reveal answer
Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:
Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).
70 + 70 + 40 equals 180 also checks out.
Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.
Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:
Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).
70 + 70 + 40 equals 180 also checks out.
Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.
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Triangle ABC has angle measures as follows:
mangle ABC=4x+3
mangle ACB=2x+6
mangle BAC=3x
What is mangle BAC?
Triangle ABC has angle measures as follows:
mangle ABC=4x+3
mangle ACB=2x+6
mangle BAC=3x
What is mangle BAC?
Tap to reveal answer
The sum of the measures of the angles of a triangle is 180.
Thus we set up the equation 4x+3+2x+6+3x=180
After combining like terms and cancelling, we have 9x=171rightarrow x=19
Thus mangle BAC=3x=57
The sum of the measures of the angles of a triangle is 180.
Thus we set up the equation 4x+3+2x+6+3x=180
After combining like terms and cancelling, we have 9x=171rightarrow x=19
Thus mangle BAC=3x=57
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The base angle of an isosceles triangle is $27^{circ}$. What is the vertex angle?
The base angle of an isosceles triangle is $27^{circ}$. What is the vertex angle?
Tap to reveal answer
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Solve the equation 27+27+x=180 for x to find the measure of the vertex angle.
x = 180 - 27 - 27
x = 126
Therefore the measure of the vertex angle is $126^{circ}$.
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Solve the equation 27+27+x=180 for x to find the measure of the vertex angle.
x = 180 - 27 - 27
x = 126
Therefore the measure of the vertex angle is $126^{circ}$.
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The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?
The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?
Tap to reveal answer
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let x = the vertex angle and 2x+5 = the base angle
So the equation to solve becomes x+(2x+5)+(2x+5)=180
Thus the vertex angle is 34 and the base angles are 73.
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let x = the vertex angle and 2x+5 = the base angle
So the equation to solve becomes x+(2x+5)+(2x+5)=180
Thus the vertex angle is 34 and the base angles are 73.
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The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?
The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?
Tap to reveal answer
Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let = vertex angle and = base angle
So the equation to solve becomes .
Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let
So the equation to solve becomes
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The base angle of an isosceles triangle is ten less than twice the vertex angle. What is the vertex angle?
The base angle of an isosceles triangle is ten less than twice the vertex angle. What is the vertex angle?
Tap to reveal answer
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let = vertex angle and = base angle
So the equation to solve becomes
So the vertex angle is 40 and the base angles is 70
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let
So the equation to solve becomes
So the vertex angle is 40 and the base angles is 70
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