To find the measures of the labeled angles in the figure, we can apply basic geometric principles such as:

1. Angles on a straight line (sum to 180°).
2. Angles in a triangle (sum to 180°).
3. Vertical angles (equal).
4. Complementary angles (sum to 90°).

Breakdown of the angles:

• The angle marked 34° is part of triangle $EHF$.
• The angle 50° is part of triangle $HFG$.

Let's proceed step by step.

1. Triangle $HFG$

In triangle $HFG$:

• The given angle is $50^\circ$.
• $a^\circ$ and $b^\circ$ are the unknowns.
• Since the sum of angles in a triangle is 180°: $a^\circ + b^\circ + 50^\circ = 180^\circ$ This gives: a^\circ + b^\circ = 130^\circ \tag{1}

2. Right Angle at $F$

At point $F$, we see there is a right angle:

• $e^\circ$ and $b^\circ$ form a 90° angle (since $FDG$ is a vertical line): e^\circ + b^\circ = 90^\circ \tag{2} From equation (2), we can solve for $e^\circ$: e^\circ = 90^\circ - b^\circ \tag{3}

3. Triangle $EHF$

In triangle $EHF$:

• The given angles are $34^\circ$ and $d^\circ$.
• Since the sum of angles in a triangle is 180°: $d^\circ + 34^\circ + c^\circ = 180^\circ$ This gives: d^\circ + c^\circ = 146^\circ \tag{4}

4. Vertical Angles at $H$

Since $c^\circ$ and the 50° angle are vertical angles, they are equal: c^\circ = 50^\circ \tag{5}

5. Solve for $d^\circ$

Using equation (4) and substituting $c^\circ = 50^\circ$: $d^\circ + 50^\circ = 146^\circ$ $d^\circ = 96^\circ$

6. Solve for $a^\circ$ and $b^\circ$

Using equation (1), substitute $a^\circ + b^\circ = 130^\circ$. We need another equation to proceed. From equation (2), solve for $b^\circ$. We already know: $e^\circ + b^\circ = 90^\circ$ Substitute equation (3) into this: $b^\circ = 90^\circ - e^\circ$ But without $e^\circ$, we would need to clarify a missing value in the image setup.

Would you like to clarify the value of $e^\circ$, or should I proceed with additional assumptions?

Relative questions:

1. How do vertical angles help in solving geometric problems?
2. What is the relationship between complementary and supplementary angles?
3. How can you verify if two angles are complementary in a right-angle triangle?
4. Why do angles in any triangle always sum to 180°?
5. How would changing the length of one triangle side affect angle measures?

Tip: Always label all known angles clearly when solving multi-angle problems to avoid confusion.