Question

what is the measure of the vertex and how do I explain measure of the vertex.

Answer 1

Answer:

122°

Explanation:

Since the triangle ABC is isosceles, the measures of angles BCA and BCA are the same. So, we can write the following equation:

$\begin{gathered} m\angle BAC=m\angle BCA \\ 7x+1=5x+9 \end{gathered}$

So, solving for x, we get:

$\begin{gathered} 7x+1-1=5x+9-1 \\ 7x=5x+8 \\ 7x-5x=5x+8-5x \\ 2x=8 \\ \frac{2x}{2}=\frac{8}{2} \\ x=4 \end{gathered}$

Therefore, the measure of the angle BAC and BCA is:

$\begin{gathered} m\angle BCA=7x+1=7(4)+1=29 \\ m\angle BAC=5x+9=5(4)+9=29 \end{gathered}$

Then, the sum of the interior angles of a triangle is 180, so the vertex angle ABC can be calculated as:

$\begin{gathered} m\angle BCA+m\angle BAC+m\angle ABC=180 \\ 29+29+m\angle ABC=180 \\ 58+m\angle ABC=180 \\ m\angle ABC=180-58 \\ m\angle ABC=122 \end{gathered}$

So, the measure of the vertex angle is 122°