To find the values of the variables for the given parallelogram, let’s analyze the information provided:

Step 1: Analyzing Angle and Side Expressions

• Angle Information: The problem provides $m\angle 1 = 5y - 15$.
• Side Lengths: There are expressions for two sides marked in red:
  • One side length is $9x + 4$
  • The other side length is $10x - 2$

Step 2: Applying Properties of Parallelograms

1. Opposite Angles are Equal: Since this is a parallelogram, opposite angles are congruent. This information might help us find the value of $y$ if we know the measure of any other angle.
2. Opposite Sides are Equal: In a parallelogram, opposite sides are equal. Therefore, we can set the expressions for the two side lengths equal to each other.

Step 3: Solving for $x$ Using Side Lengths

Since opposite sides are equal: $9x + 4 = 10x - 2$ Let's solve this equation for $x$: $9x + 4 = 10x - 2$ Subtract $9x$ from both sides: $4 = x - 2$ Add 2 to both sides: $x = 6$

Step 4: Solving for $y$ Using the Angle Measure

If additional information about the angles is provided, such as if the parallelogram is a rhombus, rectangle, or square, we could deduce specific angle measures. However, without further angle information, we assume standard properties.

If it’s implied that this is a rhombus, then adjacent angles are supplementary: $m\angle 1 + m\angle 2 = 180^\circ$

Since $m\angle 1 = 5y - 15$, we may need additional clarification on angle relationships for a complete solution.

Let me know if you'd like further details on any part of this solution, or if there's specific angle information missing.

Additional Questions:

1. How do opposite side properties change in different quadrilaterals?
2. What would change if the given shape were a rectangle instead of a parallelogram?
3. How does setting expressions equal help solve variable values in geometric figures?
4. What are supplementary angles, and why are they relevant in quadrilaterals?
5. Can properties of parallelograms apply to any quadrilateral?

Tip:

Always check for properties of geometric shapes, as they can simplify variable solutions!