PSAT Math : Acute / Obtuse Isosceles Triangles
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Example Questions
Example Question #4 : Isosceles Triangles
Triangle ABC has angle measures as follows:
What is 
19
57
44
79
90
57
The sum of the measures of the angles of a triangle is 180.
Thus we set up the equation
After combining like terms and cancelling, we have
Thus
Example Question #5 : Isosceles Triangles
The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base 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 the vertex angle is 34 and the base angles are 73.
Example Question #6 : Isosceles Triangles
The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?
Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let 
So the equation to solve becomes 
Example Question #4 : Isosceles Triangles
The base angle of an isosceles triangle is ten less than twice the vertex angle. What 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
So the vertex angle is 40 and the base angles is 70
Example Question #5 : How To Find An Angle In An Acute / Obtuse Triangle
The base angle of an isosceles triangle is 10 more than twice the vertex angle. What 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
The vertex angle is 32 degrees and the base angle is 74 degrees
Example Question #6 : How To Find An Angle In An Acute / Obtuse Triangle
In an isosceles triangle, the vertex angle is 15 less than the base angle. What is the base 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.
Example Question #11 : Isosceles Triangles
In an isosceles triangle the vertex angle is half the base angle. What 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 
Example Question #1 : Acute / Obtuse Isosceles Triangles
If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is 
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.
Example Question #2 : Acute / Obtuse Isosceles Triangles
The base angle of an isosceles triangle is 
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Solve the equation 
x = 180 - 27 - 27
x = 126
Therefore the measure of the vertex angle is 
Example Question #2 : Isosceles Triangles
Two sides of an isosceles triangle are 20 and 30. What is the difference of the largest and the smallest possible perimeters?
10
The answer cannot be determined
0
30
15
10
The trick here is that we don't know which is the repeated side. Our possible triangles are therefore 20 + 20 + 30 = 70 or 30 + 30 + 20 = 80. The difference is therefore 80 – 70 or 10.
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