A baseball diamond’s home plate has three right angles. The other two angles are congruent. Find their measure.

The theorem for the sum of interior angles for convex polygon states that the sum of measures of the interior angles of a convex polygon with sides is $\left(n-2\right)180$.

The baseball diamond’s home plate is a pentagon and the number of sides in the pentagon is 5.

Therefore, the value of n is 5.

By using the theorem for the sum of interior angles for convex polygon, the sum of the interior angles for the polygon having 5 sides is given by:

$\begin{array}{c}\left(5-2\right)180=3\times 180\\ =540\end{array}$

Therefore, the sum of interior angles for the pentagon is $540°$.

Consider one of the angle be x.

As, the two angles are congruent, therefore the other angle will also have measure x.

The sum of interior angles for the pentagon is $540°$, the measures of three angles is $90°$and the measures of the two angles is $x$.

Therefore, the equation for the given problem is:

$x+x+90+90+90=540$

Now solve the equation to find the value of $x$.

$$\begin{gathered} x+x+90+90+90=540 \\ 2x+270=540 \\ 2x=540-270 \\ 2x=270 \\ x=\frac{270}{2} \\ x=135 \end{gathered}$$

Therefore, the value of x is 135.

The value of x is 135 and the measures of the angles is .

Therefore, the measures of the angles is $135°$.