In convex pentagon $ABCDE$, angles $A$, $B$ and $C$ are congruent and angles $D$ and $E$ are congruent. If the measure of angle $A$ is 40 degrees less than the measure of angle $D$, what is the measure of angle $D$?
Let the measure of $\angle A$ be $x$, so we have $\angle B = x$ and $\angle C=x$, too. Since $\angle A$ is $40^\circ$ less than $\angle D$, we have $\angle D = x + 40^\circ$, so $\angle E = x+40^\circ$.  The sum of the angle measures in a pentagon is $180(5-2) = 540$ degrees, so we have \[x + x + x + (x+40^\circ) + (x+40^\circ) = 540^\circ.\] Simplifying the left side gives $5x + 80^\circ = 540^\circ$, so $5x = 460^\circ$ and $x = 92^\circ$.  Therefore, $\angle D = \angle A + 40^\circ = \boxed{132^\circ}$.
132
Level 5
Prealgebra
Problem: In convex pentagon $ABCDE$, angles $A$, $B$ and $C$ are congruent and angles $D$ and $E$ are congruent. If the measure of angle $A$ is 40 degrees less than the measure of angle $D$, what is the measure of angle $D$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>Let the measure of angle A be denoted as x. Since angles A, B, and C are congruent, angles B and C also measure x. Let the measure of angle D be denoted as y. Since angles D and E are congruent, angle E also measures y. The sum of the interior angles of a pentagon is given by the formula (n-2) * 180 degrees, where n is the number of sides. For a pentagon, n = 5, so the sum of the interior angles is (5-2) * 180 = 540 degrees.

Setting up the equation for the sum of the angles in pentagon ABCDE, we have:
x + x + x + y + y = 540 degrees.
This simplifies to:
3x + 2y = 540 degrees.

According to the problem, angle A is 40 degrees less than angle D, which gives us the relationship:
x = y - 40 degrees.

Substituting this expression for x into the angle sum equation:
3(y - 40) + 2y = 540.
Expanding this, we get:
3y - 120 + 2y = 540.
Combining like terms results in:
5y - 120 = 540.

Adding 120 to both sides yields:
5y = 660.

Dividing both sides by 5 gives:
y = 132 degrees.

Thus, the measure of angle D is 132 degrees.<|END_OF_LATENT|> The final answer is $132$.