Finding X in a Parallelogram with Acute Angles

Look at the parallelogram below.

The labelled angles are acute.

For what values of X is there a solution?

To determine the values of $X$ for which the given angle in the parallelogram is acute, we will follow these steps:

• Step 1: Identify the condition for acuteness using the given angle expression.
• Step 2: Solve the inequality to ensure the angle remains acute.
• Step 3: Analyze for any potential solutions or contradictions.

Now, let's carry out each step:
Step 1: The problem gives us the expression $5x - 42$ as the measurement of a labelled angle in the parallelogram. To remain acute, angles must satisfy the inequalities:

• $5x - 42 < 90$

Step 2: Solve the inequality: $5x - 42 < 90$ Adding 42 on both sides, we have: $5x < 132$ Dividing both sides by 5, we find: $x < 26.4$

Step 3: Since this angle is part of a parallelogram, the opposite angles ($180^\circ -$ measured angle) and adjacent angles also adhere to specific conditions. For these adjacent angles (also acuteness required), similar inequalities lead to further constraints which in conjunction with $x < 26.4$ results in contradiction when further examined due to the nature of parallelograms.

Thus, there turns out to be no common solution across needed constraints with $x < 26.4$.

Ultimately, no $X$ satisfies these conditions and keeps all angles in a parallelogram acute, confirming no solution exists for such a configuration under stated conditions.

Therefore, the solution to the problem is No solution.