All triangles have degrees.  An isoceles triangle has one vertex angle and two congruent base angles.

Let vertex angle and  base angle.

So the equation to solve becomes:

or

Thus for the vertex angle and for the base angle.

The sum of the vertex and one base angle is .

Every triangle has degrees.  An isoceles triangle has one vertex angle and two congruent base angles.

Let base angle and vertex angle.

So the equation to solve becomes .

Thus the base angles are and the vertex angle is .

Every triangle has 180 degrees.  An isoceles triangle has one vertex angle and two congruent base angles.

Let vertex angle and  base angle.

Then the equation to solve becomes:

, or .

Then the vertex angle is , the base angle is , and the product is .

Because the interior angles of a triangle add up to 180°, we can create an equation using the variables given in the problem: x+2x+(3x+30)=180. This simplifies to 6X+30=180. When we subtract 30 from both sides, we get 6x=150. Then, when we divide both sides by 6, we get x=25. Because Angle B=2x degrees, we multiply 25 times 2. Thus, Angle B is equal to 50°. If you got an answer of 25, you may have forgotten to multiply by 2. If you got 105, you may have found Angle C instead of Angle B.

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let  = vertex angle and  = base angle.

Then the equation to solve becomes

or

.

Solving for  gives a vertex angle of 24 degrees and a base angle of 78 degrees.

Every triangle has . An isosceles triangle has one vertex ange, and two congruent base angles.

Let  be the vertex angle and  be the base angle.

The equation to solve becomes , since the base angle occurs twice.

Now we can solve for the vertex angle.

The difference between the vertex angle and the base angle is .

Each triangle has degrees.

An isoceles triangle has two congruent base angles and one vertex angle.

Let vertex angle and base angle.

Then the equation to solve becomes or .

Add to both sides to get .

Divide both sides by to get vertex angle and base angles, so the sum of the angles is .

Every triangle contains degrees.  An isoceles triangle has two congruent base angles and one vertex angle.

Let the vertex angle and the base angle

So the equation to solve becomes or and dividing by gives for the vertex angle and for the base angle, so the sum is 

The answer is .

Since we know that the permieter is  inches and one side is  inches, it can be determined that the remaining two sides must combine to be  inches. The ratio of the remaining two sides is  which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation , and divide both sides by  which means . The ratio of the remaining side lengths then becomes  or . We now know the 3 side lengths are .

 is the shortest side and thus the answer.