The sum of the measures of the angles of a triangle is 180.

Thus we set up the equation \(\displaystyle \dpi{100} \small 4x+3+2x+6+3x=180\)

After combining like terms and cancelling, we have \(\displaystyle \dpi{100} \small 9x=171\rightarrow x=19\)

Thus \(\displaystyle \dpi{100} \small m\angle BAC=3x=57\)

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

Let \(\displaystyle x\) = the vertex angle and \(\displaystyle 2x+5\) = the base angle

So the equation to solve becomes  \(\displaystyle x+(2x+5)+(2x+5)=180\)

Thus the vertex angle is 34 and the base angles are 73.

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

Let \(\displaystyle x\) = vertex angle and \(\displaystyle 3x-15\) = base angle

So the equation to solve becomes \(\displaystyle x+(3x-15)+(3x-15)=180\).

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

Let \(\displaystyle x\) = vertex angle and \(\displaystyle 2x - 10\) = base angle

So the equation to solve becomes  \(\displaystyle x + (2x - 10) + (2x - 10) = 180\)

So the vertex angle is 40 and the base angles is 70

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

Let \(\displaystyle x\)= the vertex angle and \(\displaystyle 2x + 10\) = the base angle

So the equation to solve becomes \(\displaystyle x + (2x +10) + (2x +10) = 180\)

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 \(\displaystyle x\) = base angle and \(\displaystyle x - 15\) = vertex angle

So the equation to solve becomes \(\displaystyle (x - 15) + x + x = 180\)

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 \(\displaystyle x\) = base angle and \(\displaystyle 0.5x\) = vertex angle

So the equation to solve becomes \(\displaystyle x+x+0.5x=180\), thus \(\displaystyle x=72\) is the base angle and \(\displaystyle 0.5x=36\) 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:

\(\displaystyle \frac{x+y}{2}=55^o\)

\(\displaystyle x+x+y=180^o\)

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 \(\displaystyle 27+27+x=180\) for x to find the measure of the vertex angle. 

x = 180 - 27 - 27

x = 126

Therefore the measure of the vertex angle is \(\displaystyle 126^{\circ}\).

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.