To solve for $y$ in this parallelogram, we can use the properties of parallelograms and angle relationships.

1. Identify opposite angles: In a parallelogram, opposite angles are equal. Therefore, the angle $(5y)^\circ$ at $D$ is equal to the angle $(7x - 5)^\circ$ at $F$.

2. Sum of adjacent angles: Adjacent angles in a parallelogram are supplementary (they add up to $180^\circ$). So, at point $Y$, the sum of the angles $45^\circ$, $70^\circ$, and the angle at $E$ should be $180^\circ$.

Let's proceed with the calculations:

Step 1: Find the angle at $E$ using supplementary angles

The sum of angles $45^\circ$ and $70^\circ$ is: $45^\circ + 70^\circ = 115^\circ$ Thus, the angle at $E$ is: $180^\circ - 115^\circ = 65^\circ$

Step 2: Set up an equation for $5y$

Since $(5y)^\circ$ is also an opposite angle to $65^\circ$, we have: $5y = 65$ $y = \frac{65}{5} = 13$

Final Answer

$y = 13$

Would you like more detailed explanations on any step?

Here are five related questions you might find interesting:

1. How would we solve for $x$ if given a similar angle property?
2. What are other properties of parallelograms that are useful in geometry?
3. How do adjacent and opposite angles behave in other quadrilaterals like rectangles or rhombuses?
4. Can this approach be applied to other types of polygons?
5. How can we confirm that a quadrilateral is a parallelogram based on its angles?

Tip: In geometry, identifying and using symmetrical properties like those in parallelograms often simplifies complex angle calculations.