lecture 5. parallel lines (problem set)
9 - 10. for each of the following, list all pairs of alternate interior angles and corresponding angles, if there are none, then list all pairs of interior angles on the same side of the transversal. indicate the parallel lines which form each pair of angles.

Question

Accepted Answer

# Explanation:
## Step1: Recall angle - pair definitions
Alternate interior angles are non - adjacent angles between two parallel lines and on opposite sides of a transversal. Corresponding angles are in the same relative position with respect to the parallel lines and the transversal. Interior angles on the same side of the transversal are between the parallel lines and on the same side of the transversal.
## Step2: Analyze the first figure (left - hand side)
Parallel lines are $AB\parallel CD$ and $AD\parallel BC$.
For the transversal $BD$:
- Alternate interior angles: $\angle ADB$ and $\angle DBC$, $\angle CDB$ and $\angle ABD$ (formed by $AD\parallel BC$).
- There are no corresponding angles with respect to the given parallel lines and transversal $BD$.
- Interior angles on the same side of the transversal $BD$: None relevant in this case for the given parallel - line pairs.
For the transversals $AC$ (not shown in the problem but for completeness), if we consider $AB\parallel CD$ and $AD\parallel BC$:
- Alternate interior angles: $\angle BAC$ and $\angle ACD$, $\angle CAD$ and $\angle ACB$ (formed by $AB\parallel CD$ and $AD\parallel BC$ respectively).
- No corresponding angles for the given parallel lines and transversal $AC$.
- Interior angles on the same side of the transversal $AC$: None relevant for the given parallel - line pairs.
## Step3: Analyze the second figure (right - hand side)
Parallel lines are $AB\parallel CD$ and $AD\parallel BC$.
For the transversal $BD$:
- Alternate interior angles: $\angle ADB$ and $\angle DBC$, $\angle CDB$ and $\angle ABD$ (formed by $AD\parallel BC$).
- For the transversal $AC$:
  - Alternate interior angles: $\angle BAC$ and $\angle ACD$, $\angle CAD$ and $\angle ACB$ (formed by $AB\parallel CD$ and $AD\parallel BC$ respectively).
  - There are no corresponding angles with respect to the given parallel lines and transversals $AC$ and $BD$.
  - Interior angles on the same side of the transversal: None relevant for the given parallel - line pairs and transversals in a non - overlapping way.

# Answer:
**First figure**:
- Alternate interior angles with respect to $BD$ (formed by $AD\parallel BC$): $\angle ADB$ and $\angle DBC$, $\angle CDB$ and $\angle ABD$; with respect to $AC$ (formed by $AB\parallel CD$ and $AD\parallel BC$): $\angle BAC$ and $\angle ACD$, $\angle CAD$ and $\angle ACB$.
- Corresponding angles: None.
- Interior angles on the same side of the transversal: None.

**Second figure**:
- Alternate interior angles with respect to $BD$ (formed by $AD\parallel BC$): $\angle ADB$ and $\angle DBC$, $\angle CDB$ and $\angle ABD$; with respect to $AC$ (formed by $AB\parallel CD$ and $AD\parallel BC$): $\angle BAC$ and $\angle ACD$, $\angle CAD$ and $\angle ACB$.
- Corresponding angles: None.
- Interior angles on the same side of the transversal: None.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall angle - pair definitions

Step2: Analyze the first figure (left - hand side)

Step3: Analyze the second figure (right - hand side)

Answer: