We are given a triangle with two of its angles known to be $42^\circ$ and $63^\circ$. We need to find the measure of the third angle, which is labeled as A in the figure.

1. Problem Description

We are given a triangle with two of its angles known to be $42^\circ$ and $63^\circ$. We need to find the measure of the third angle, which is labeled as A in the figure.

2. Solution Steps

The sum of the angles in a triangle is always $180^\circ$. Therefore, if we know two angles, we can find the third angle by subtracting the sum of the two known angles from $180^\circ$.

Let the three angles of the triangle be $\alpha$, $\beta$, and $\gamma$. In our case, we have $\alpha = 42^\circ$ and $\beta = 63^\circ$. We want to find $\gamma$.

The sum of angles in a triangle is:

$\alpha + \beta + \gamma = 180^\circ$

$42^\circ + 63^\circ + \gamma = 180^\circ$

$105^\circ + \gamma = 180^\circ$

$\gamma = 180^\circ - 105^\circ$

$\gamma = 75^\circ$

3. Final Answer

The measure of the third angle A is $75^\circ$.