https://file.aiquickdraw.com/mathbot/file/1726527212786c9nxp08o.jpg

Let's break down the diagram to solve for the indicated angles based on the information provided.

We have several lines intersecting, and some angle measures are given:
- Angle \(142^\circ\) is provided, along with \(53^\circ\).
- There's a right angle (\(90^\circ\)) indicated as well.

Let's go step by step:

### Step 1: Identifying angles

#### a) \(a^\circ\):
- Since \(a^\circ\) and \(142^\circ\) form a linear pair, they are supplementary angles. Therefore:
  \[
  a^\circ = 180^\circ - 142^\circ = 38^\circ
  \]

#### b) \(b^\circ\):
- \(b^\circ\) is vertically opposite to \(142^\circ\). Vertically opposite angles are equal, so:
  \[
  b^\circ = 142^\circ
  \]

#### c) \(c^\circ\):
- \(c^\circ\) is adjacent to \(a^\circ\) and they form a linear pair (on the straight line), so:
  \[
  c^\circ = 180^\circ - 38^\circ = 142^\circ
  \]

#### d) \(d^\circ\):
- \(d^\circ\) is vertically opposite to the \(53^\circ\) angle. Thus, they are equal:
  \[
  d^\circ = 53^\circ
  \]

#### e) \(e^\circ\):
- \(e^\circ\) forms a right angle (perpendicular lines), so:
  \[
  e^\circ = 90^\circ
  \]

#### f) \(f^\circ\):
- \(f^\circ\) and \(e^\circ\) form a linear pair on the same straight line, so:
  \[
  f^\circ = 180^\circ - 90^\circ = 90^\circ
  \]

### Final Answers:
- a) \(38^\circ\)
- b) \(142^\circ\)
- c) \(142^\circ\)
- d) \(53^\circ\)
- e) \(90^\circ\)
- f) \(90^\circ\)

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the relationship between vertically opposite angles?
2. How do you calculate supplementary angles?
3. What defines a linear pair of angles?
4. Can two adjacent angles be both acute? Why or why not?
5. How do perpendicular lines affect the calculation of angles?

**Tip:** Vertically opposite angles are always equal, which simplifies many angle problems in geometry.

Given the diagram, find the indicated angle measures.

Solve for the indicated angles in a geometry problem involving vertically opposite angles, linear pairs, and perpendicular angles. The problem provides angle measures of 142° and 53°, leading to a step-by-step solution suitable for students in Grades 8-10.

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