In an isosceles triangle the measure of the vertex angle is (x+3)^circ . The measures of the b are (3x+1)^circ and (4y+4)^circ , respectively. What is the measure of the vertex angle of the isosceles triangle? A. 28^circ B. 47^circ C. 76^circ D. 104^circ

## Step 1: Setting up the equation<br />### Since the triangle is isosceles, two of its angles must be equal. We are given that the base angles are $(3x+1)^{\circ}$ and $(4y+4)^{\circ}$. Therefore, we can set up the equation $3x+1 = 4y+4$.<br /><br />## Step 2: Solving for x in terms of y<br />### Subtract 1 from both sides of the equation: $3x = 4y+3$. Then, divide both sides by 3: $x = \frac{4y+3}{3}$.<br /><br />## Step 3: Using the triangle sum theorem<br />### The sum of the angles in a triangle is $180^{\circ}$. So, $(x+3) + (3x+1) + (4y+4) = 180$. Simplifying, we get $4x + 4y + 8 = 180$, or $4x + 4y = 172$. Dividing by 4, we get $x + y = 43$.<br /><br />## Step 4: Substituting x and solving for y<br />### Substitute $x = \frac{4y+3}{3}$ into $x + y = 43$: $\frac{4y+3}{3} + y = 43$. Multiply both sides by 3: $4y+3 + 3y = 129$. Combine like terms: $7y + 3 = 129$. Subtract 3 from both sides: $7y = 126$. Divide by 7: $y = 18$.<br /><br />## Step 5: Solving for x<br />### Substitute $y=18$ into $x+y=43$: $x + 18 = 43$. Subtract 18 from both sides: $x = 25$.<br /><br />## Step 6: Finding the vertex angle<br />### The vertex angle is $(x+3)^{\circ}$. Substitute $x=25$: $25+3 = 28$.