Solve for X: Trapezoid with Perimeter 26 and Side Length X+1

To solve the problem, we will calculate the perimeter using the expression for each side:

• The top base is $10$.
• The bottom base is $6 + X$.
• One leg is $X$.
• The other leg is $X + 1$.

According to the problem, the perimeter of the trapezoid is given as $26$.

Let's write the equation for the perimeter:

$10 + (6 + X) + X + (X + 1) = 26$

Simplify the expression:

$10 + 6 + X + X + X + 1 = 26$

This simplifies to:

$17 + 3X = 26$

Subtract $17$ from both sides of the equation:

$3X = 26 - 17$

Simplify the right side:

$3X = 9$

Divide both sides by $3$ to solve for $X$:

$X = \frac{9}{3} = 3$

Therefore, the value of $X$ is $\mathbf{3}$.

Substitute $X = 3$ back into the dimensions to verify:

$10 + (6 + 3) + 3 + (3 + 1) = 26$

$10 + 9 + 3 + 4 = 26$

The calculation confirms the given perimeter of $26$, verifying our solution. Thus, the correct value of $X$ is $\mathbf{3}$.