Let’s solve each part step by step:

(a) Find $m\angle M$:

• The triangle $\triangle PNM$ is isosceles because the markings indicate that $PN = PM$. In an isosceles triangle, the base angles are congruent. This means $m\angle N = m\angle M$.
• Since the sum of angles in a triangle is $180^\circ$, we have: $m\angle P + m\angle N + m\angle M = 180^\circ$
• Given that $m\angle N = m\angle M$, let $x = m\angle M$. Then: $m\angle P + 2x = 180^\circ$
• Since no specific measure is given for $m\angle P$, its value is needed to solve $x$.

Can you confirm the measure of $m\angle P$, or do you want to proceed assuming $m\angle P = 60^\circ$ (a common case in similar problems)?

(b) Find $m\angle Q$:

• In $\triangle QRS$, we know:

  • $m\angle R = 100^\circ$
  • $QR = RS$ (isosceles triangle).

• Since $QR = RS$, the base angles $m\angle Q$ and $m\angle S$ are congruent. Let $x = m\angle Q$. Then: $m\angle Q + m\angle S + m\angle R = 180^\circ$ $x + x + 100^\circ = 180^\circ$ $2x = 80^\circ$ $x = 40^\circ$

• So, $m\angle Q = 40^\circ$.

Summary of Solutions:

1. (a): Need clarification on $m\angle P$ or further assumptions.
2. (b): $m\angle Q = 40^\circ$.

Would you like more details or have further questions?

Related Questions:

1. How do you calculate the third angle in a triangle given two angles?
2. What properties of isosceles triangles help in solving for unknown angles?
3. Can an isosceles triangle have an angle greater than $90^\circ$?
4. Why does the sum of the angles in any triangle always equal $180^\circ$?
5. How do markings on a triangle indicate equal sides or angles?

Tip:

Always check the markings on a triangle to determine congruence relations—they provide valuable shortcuts for solving problems!