https://file.aiquickdraw.com/mathbot/file/1727915775177trxfwkzo.jpg

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.

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

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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.

Find the measures of the angles labeled in the figure on the right.

Solve for the unknown angles in the given geometry figure, using the triangle sum theorem, vertical angles, and complementary angles concepts. This problem is ideal for students in Grades 8-10.

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