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  = 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  = base angle and  = vertex angle

So the equation to solve becomes , thus  is the base angle and  is the vertex angle.

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.

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 .

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.

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.

The triangle has two sides of equal length, 13, so it is by definition isosceles. 

To determine whether the triangle is acute, right, or obtuse, compare the sum of the squares of the lengths of the two shortest sides to the square of the length of the longest side. The former quantity is equal to

The latter quantity is equal to

The former is greater than the latter; consequently, the triangle is acute. The correct response is that the triangle is acute and isosceles.