All PSAT Math Resources
Example Questions
Example Question #1 : How To Find The Perimeter Of A Polygon
Note: Figure NOT drawn to scale.
Refer to the above figure. The white trapezoid is isosceles. Give the perimeter of the blue polygon.
The lower base of this trapezoid is units longer than the upper base, and being isosceles, it is symmetrical. As a result, the lower leg of the right triangle at bottom is half this difference, or 18, which is the same as the upper leg. That makes the right triangle isosceles, and, therefore, a 45-45-90 triangle. Subsequently, the hypotenuse, which is one leg of the trapezoid, has length times a leg of the triangle, or .
The blue polygon has two sides that are the legs of this isosceles trapezoid, both of which have length ; its other three sides are of length 64, 50, 100, and 50. Add them:
Example Question #431 : Psat Mathematics
Triangle ABC has angle measures as follows:
What is ?
79
19
44
57
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 #431 : Psat Mathematics
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 = the vertex angle and = the base angle
So the equation to solve becomes
Thus the vertex angle is 34 and the base angles are 73.
Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle
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 = vertex angle and = base angle
So the equation to solve becomes .
Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle
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 = vertex angle and = base angle
So the equation to solve becomes
So the vertex angle is 40 and the base angles is 70
Example Question #141 : Triangles
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 = 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
Example Question #432 : Geometry
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 = base angle and = vertex angle
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 = base angle and = vertex angle
So the equation to solve becomes , thus is the base angle and is the vertex angle.
Example Question #431 : Psat Mathematics
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?
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 : Isosceles Triangles
The base angle of an isosceles triangle is . What is the vertex angle?
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Solve the equation for x to find the measure of the vertex angle.
x = 180 - 27 - 27
x = 126
Therefore the measure of the vertex angle is .
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