Let's solve the three problems step by step.

Problem (a):

This is a set of intersecting lines where $a$ and $b$ are the unknown angles. The given angle is $75^\circ$.

• Observations:

  • Since the two lines intersect, $a$ and $75^\circ$ are vertically opposite angles, meaning $a = 75^\circ$.
  • Angles $b$ and $75^\circ$ form a linear pair, meaning they are supplementary ($b + 75^\circ = 180^\circ$).

• Solution: $a = 75^\circ$ $b = 180^\circ - 75^\circ = 105^\circ$

Problem (b):

This is a triangle with angles $55^\circ, 73^\circ,$ and $c$ as the unknown angle.

• Observations:

  • The sum of angles in a triangle is always $180^\circ$.
  • Thus, $55^\circ + 73^\circ + c = 180^\circ$.

• Solution: $c = 180^\circ - (55^\circ + 73^\circ) = 180^\circ - 128^\circ = 52^\circ$

Problem (c):

This is a triangle with one given angle $55^\circ$ and two unknown angles $a$ and $b$. One corner is marked as a right angle ($90^\circ$).

• Observations:

  • The sum of angles in a triangle is $180^\circ$.
  • One angle is $90^\circ$, so $90^\circ + 55^\circ + b = 180^\circ$.
  • Solving for $b$, we then use $a$ as part of the linear pair with $50^\circ$ (external angle property).

• Solution for $b$: $b = 180^\circ - (90^\circ + 55^\circ) = 180^\circ - 145^\circ = 35^\circ$

• Solution for $a$: From the external angle property, $a = 50^\circ$.

Final Answers:

• (a) $a = 75^\circ, b = 105^\circ$
• (b) $c = 52^\circ$
• (c) $a = 50^\circ, b = 35^\circ$

Would you like detailed explanations or have further questions? Here are 5 questions to expand your understanding:

1. What is the vertical angle property, and how does it work in problem (a)?
2. Can you prove the sum of angles in a triangle is always $180^\circ$?
3. How does the external angle property apply to problem (c)?
4. Why are linear pairs always supplementary?
5. How would you solve similar problems with more unknowns?

Tip: Always identify angle properties (linear pairs, vertical angles, etc.) to simplify geometry problems!