Let's analyze the diagram to calculate each numbered angle.

Key observations:

• The triangle $DEF$ has isosceles sides $DE$ and $DF$ as marked with congruent lines (6 and 7), which helps deduce symmetry in angles.
• $ABGF$ is a quadrilateral with a right angle at $A$ and $F$, and we have given angle measures at $F$ and $G$.
• Some angles are connected to parallel lines due to the shape of the quadrilateral and the transversal crossing the parallel lines.

Let's go through each numbered angle:

1. Angle 1 (at $A$)

• $\angle A$ forms part of a right angle in quadrilateral $ABGF$.
• Since quadrilateral $ABGF$ contains two right angles at $A$ and $F$, we know $\angle 1 = 90^\circ$.

2. Angle 2 (at $B$)

• $\angle 2$ is adjacent to angle 1 and together they form a straight line at $B$. So: $\angle 1 + \angle 2 = 180^\circ$ $\angle 2 = 180^\circ - 90^\circ = 90^\circ$

3. Angle 3 (at $B$, part of triangle $BCF$)

• $\angle 3$ forms part of triangle $BCF$.
• We know that $\angle 4 = 32^\circ$ and $\angle F = 90^\circ$.
• Using the triangle angle sum property: $\angle 3 + \angle 4 + \angle F = 180^\circ$ $\angle 3 + 32^\circ + 90^\circ = 180^\circ$ $\angle 3 = 180^\circ - 122^\circ = 58^\circ$

4. Angle 4 (given at $C$)

• $\angle 4$ is already given as $32^\circ$.

5. Angle 5 (at $C$, between triangles $DCE$ and $DCF$)

• Since triangle $DCE$ is isosceles ($DE = DF$), angles at $D$ and $E$ are equal. Let’s denote them as $x$.
• Using the fact that the sum of angles in a triangle equals 180°: $x + x + 32^\circ = 180^\circ$ $2x = 148^\circ$ $x = 74^\circ$
• Therefore, angle 5 is $74^\circ$.

To summarize:

• $\angle 1 = 90^\circ$
• $\angle 2 = 90^\circ$
• $\angle 3 = 58^\circ$
• $\angle 4 = 32^\circ$
• $\angle 5 = 74^\circ$

Would you like more details on any specific angle? Here are some related questions to expand on this:

1. Why is angle 3 calculated based on the triangle angle sum property?
2. How does the isosceles triangle property help calculate angle 5?
3. What are alternate interior angles, and how might they help with parallel lines?
4. Can we verify the angle measures by calculating the interior angles of quadrilateral $ABGF$?
5. Are there any other geometric properties visible in this diagram (e.g., congruence)?

Tip: When solving geometric problems, always look for congruent sides and angles to simplify your work.