To solve for the measure of angle $A$ in the triangle, we’ll use the fact that the sum of the interior angles of any triangle is $180^\circ$.

Given:

• $\angle A = 6x$
• $\angle B = 2x + 5$
• $\angle C = 7x - 5$

Step 1: Set up the equation

Since the sum of the angles in a triangle is $180^\circ$, we can write: $6x + (2x + 5) + (7x - 5) = 180$

Step 2: Simplify the equation

Combine like terms: $6x + 2x + 7x + 5 - 5 = 180$ $15x = 180$

Step 3: Solve for $x$

$x = \frac{180}{15} = 12$

Step 4: Substitute $x = 12$ into $\angle A$

Since $\angle A = 6x$: $\angle A = 6 \times 12 = 72^\circ$

Answer

The measure of angle $A$ is $72^\circ$.

Would you like more details on any part of the solution?

Here are some related questions for further practice:

1. How would you solve for the measures of angles $B$ and $C$ in this triangle?
2. If the angle measures are changed, how would you find a new value for $x$?
3. How would you set up an equation if you knew two of the angles and needed to find the third?
4. Can a triangle have an angle measure greater than $90^\circ$ if it's scalene?
5. How would you verify your answer using the angle sum property?

Tip: Always verify your results by substituting values back into the original conditions. This helps confirm accuracy.