Because line segments ZA and ZB are radii of the circle, they must have the same length. That makes triangle ABZ an isosceles triangle, with <ZAB and <ZBA having the same measure. Because the three angles of a triangle must sum to 180°, you can express this in the equation:

140 + 2x = 180 --> 2x = 40 --> x = 20

It's good to draw a diagram for this; we know that it's an isosceles triangle; remember that the angles of a triangle total 180 degrees.

Angle G for this triangle is the one angle that doesn't correspond to an equal side of the isosceles triangle (opposite side to the angle), so that means ∠F = ∠H, and that ∠F + ∠H + 40 = 180,

By substitution we find that ∠F * 2 = 140 and angle F = 70 degrees. 

An isosceles triangle has two congruent base angles and one vertex angle.  Each triangle contains \(\displaystyle 180^{\circ}\).  Let \(\displaystyle x\) = base angle, so the equation becomes \(\displaystyle 54 + x + x = 180\).  Solving for \(\displaystyle x\) gives \(\displaystyle x = 63\)

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

Let \(\displaystyle x\) = the vertex angle

and \(\displaystyle 2x - 5\) = base angle

So the equation to solve becomes 

\(\displaystyle x + (2x - 5) + (2x - 5) = 180\)

or

\(\displaystyle 5x - 10 = 180\)

Thus the vertex angle is 38 and the base angle is 71 and their sum is 109.

This triangle has an angle of \(\displaystyle 65^{\circ}\). We also know it has another angle of \(\displaystyle 65^{\circ}\) at \(\displaystyle \angle ABC\) because the two sides are equal. Adding those two angles together gives us \(\displaystyle 130^{\circ}\) total. Since a triangle has \(\displaystyle 180^{\circ}\) total, we subtract 130 from 180 and get 50.

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

Let \(\displaystyle x\) = vertex angle and \(\displaystyle 3x + 6\) = base angle.

Then the equation to solve becomes

 \(\displaystyle x + (3x + 6) + (3x + 6) = 180\)

or

\(\displaystyle 7x + 12 = 180\).

Solving for \(\displaystyle x\) gives a vertex angle of 24 degrees and a base angle of 78 degrees.

Every triangle has \(\displaystyle 180^o\). An isosceles triangle has one vertex ange, and two congruent base angles.

Let \(\displaystyle x\) be the vertex angle and \(\displaystyle 3x + 13\) be the base angle.

The equation to solve becomes \(\displaystyle x + (3x + 13) + (3x + 13) = 180^o\), since the base angle occurs twice.

\(\displaystyle 7x + 26 = 180^o\)

\(\displaystyle 7x=154^o\)

\(\displaystyle x=22^o\)

Now we can solve for the vertex angle.

\(\displaystyle 3x+13=3(22)+13=79^o\)

The difference between the vertex angle and the base angle is \(\displaystyle 79^o - 22^o = 57^o\).

By definition, an acute isosceles triangle will have at least two sides (and at least two corresponding angles) that are congruent, and no angle will be greater than \(\displaystyle 90^{\circ}\). Addtionally, like all triangles, the three angles will sum to \(\displaystyle 180^{\circ}\). Thus, of our two answers which sum to \(\displaystyle 180^{\circ}\), only \(\displaystyle 70^{\circ}, 70^{\circ}\) is valid, as \(\displaystyle 40^{\circ}, 100^{\circ}\) would violate the "acute" part of the definition.

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.

The answer is \(\displaystyle 12\).

Since we know that the permieter is \(\displaystyle 40\) inches and one side is \(\displaystyle 12\) inches, it can be determined that the remaining two sides must combine to be \(\displaystyle 40-12=28\) inches. The ratio of the remaining two sides is \(\displaystyle 3:4\) which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation \(\displaystyle 7x=28\), and divide both sides by \(\displaystyle 7\) which means \(\displaystyle x=4\). The ratio of the remaining side lengths then becomes \(\displaystyle 3*(4):4*(4)\) or \(\displaystyle 12:16\). We now know the 3 side lengths are \(\displaystyle 12,12,and\ 16\).

\(\displaystyle 12\) is the shortest side and thus the answer.