In this diagram, we need to find the values of $c$ and $d$. The angles given involve expressions in terms of $c$ and $d$, along with known angles. To solve this, we can rely on the fact that the sum of angles in a triangle is always $180^\circ$.

Step-by-step solution:

1. Identify the triangle and its angles:
   The triangle formed has the following angles:

   • One angle is $60^\circ$,
   • The second angle is $2c - 15^\circ$,
   • The third angle is $2d^\circ$.

2. Set up an equation using the triangle sum rule:
   The sum of the interior angles of a triangle is $180^\circ$. Therefore, the equation is: $60^\circ + (2c - 15^\circ) + 2d^\circ = 180^\circ$ Simplifying this: $45^\circ + 2c + 2d = 180^\circ$ $2c + 2d = 135^\circ$ $c + d = 67.5^\circ \quad \text{(Equation 1)}$

3. Examine the parallelogram angle relationships:
   In the parallelogram, opposite angles are supplementary (i.e., their sum is $180^\circ$). Therefore, we have the equation for the supplementary angle pair: $(c + 55^\circ) + (2d^\circ) = 180^\circ$ Simplifying this: $c + 2d + 55^\circ = 180^\circ$ $c + 2d = 125^\circ \quad \text{(Equation 2)}$

4. Solve the system of equations:
   Now we have two equations:

   1. $c + d = 67.5^\circ$
   2. $c + 2d = 125^\circ$

   Subtract the first equation from the second: $(c + 2d) - (c + d) = 125^\circ - 67.5^\circ$ $d = 57.5^\circ$

   Substitute $d = 57.5^\circ$ into Equation 1: $c + 57.5^\circ = 67.5^\circ$ $c = 10^\circ$

Final values:

• $c = 10^\circ$
• $d = 57.5^\circ$

Would you like further details on the steps?

Related Questions:

1. How do you derive angle relationships in parallelograms?
2. What is the sum of interior angles in different polygons?
3. How does the triangle sum property work in other types of triangles?
4. Can we apply the same reasoning to quadrilaterals like trapezoids?
5. What are alternate ways to solve systems of linear equations?

Tip:

When solving geometry problems, always check whether certain shapes (like parallelograms or triangles) offer specific angle properties that can simplify your work!