Question

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.

Explanation:

Step1: Recall angle - line relationships

When two parallel lines are cut by a transversal, we can identify different types of angles. In a parallelogram \(ABCD\) with \(AB\parallel CD\) and \(AD\parallel BC\).

Step2: Identify alternate - interior angles

For the first parallelogram:

• Alternate - interior angles:
• When \(AB\parallel CD\) and \(AC\) is the transversal, \(\angle BAC\) and \(\angle ACD\), \(\angle BCA\) and \(\angle CAD\). When \(AD\parallel BC\) and \(AC\) is the transversal, \(\angle DAC\) and \(\angle BCA\), \(\angle BAC\) and \(\angle ACD\). When \(AB\parallel CD\) and \(BD\) is the transversal, \(\angle ABD\) and \(\angle BDC\), \(\angle ADB\) and \(\angle DBC\). When \(AD\parallel BC\) and \(BD\) is the transversal, \(\angle ADB\) and \(\angle DBC\), \(\angle ABD\) and \(\angle BDC\).
• Corresponding angles: There are no corresponding - angle pairs in a parallelogram formed by its sides as parallel lines and its diagonals as transversals in the traditional sense (since corresponding angles are defined for a single transversal intersecting two parallel lines in a non - closed - loop configuration).
• Interior angles on the same side of the transversal:
• When \(AB\parallel CD\) and \(AC\) is the transversal, \(\angle BAC\) and \(\angle ACD\) (also alternate - interior), \(\angle BCA\) and \(\angle CAD\) (also alternate - interior). When \(AD\parallel BC\) and \(AC\) is the transversal, \(\angle DAC\) and \(\angle BCA\), \(\angle BAC\) and \(\angle ACD\). When \(AB\parallel CD\) and \(BD\) is the transversal, \(\angle ABD\) and \(\angle BDC\), \(\angle ADB\) and \(\angle DBC\). When \(AD\parallel BC\) and \(BD\) is the transversal, \(\angle ADB\) and \(\angle DBC\), \(\angle ABD\) and \(\angle BDC\).

For the second parallelogram (the same analysis as above):

• Alternate - interior angles:
• When \(AB\parallel CD\) and \(AC\) is the transversal: \(\angle BAC\) and \(\angle ACD\), \(\angle BCA\) and \(\angle CAD\). When \(AD\parallel BC\) and \(AC\) is the transversal: \(\angle DAC\) and \(\angle BCA\), \(\angle BAC\) and \(\angle ACD\). When \(AB\parallel CD\) and \(BD\) is the transversal: \(\angle ABD\) and \(\angle BDC\), \(\angle ADB\) and \(\angle DBC\). When \(AD\parallel BC\) and \(BD\) is the transversal: \(\angle ADB\) and \(\angle DBC\), \(\angle ABD\) and \(\angle BDC\).
• Corresponding angles: None.
• Interior angles on the same side of the transversal:
• When \(AB\parallel CD\) and \(AC\) is the transversal: \(\angle BAC\) and \(\angle ACD\), \(\angle BCA\) and \(\angle CAD\). When \(AD\parallel BC\) and \(AC\) is the transversal: \(\angle DAC\) and \(\angle BCA\), \(\angle BAC\) and \(\angle ACD\). When \(AB\parallel CD\) and \(BD\) is the transversal: \(\angle ABD\) and \(\angle BDC\), \(\angle ADB\) and \(\angle DBC\). When \(AD\parallel BC\) and \(BD\) is the transversal: \(\angle ADB\) and \(\angle DBC\), \(\angle ABD\) and \(\angle BDC\).

Answer:

For both parallelograms:

• Alternate - interior angles:
• When \(AB\parallel CD\) and \(AC\) is the transversal: \(\angle BAC\) and \(\angle ACD\), \(\angle BCA\) and \(\angle CAD\).
• When \(AD\parallel BC\) and \(AC\) is the transversal: \(\angle DAC\) and \(\angle BCA\), \(\angle BAC\) and \(\angle ACD\).
• When \(AB\parallel CD\) and \(BD\) is the transversal: \(\angle ABD\) and \(\angle BDC\), \(\angle ADB\) and \(\angle DBC\).
• When \(AD\parallel BC\) and \(BD\) is the transversal: \(\angle ADB\) and \(\angle DBC\), \(\angle ABD\) and \(\angle BDC\).
• Corresponding angles: None.
• Interior angles on the same side of the transversal:
• When \(AB\parallel CD\) and \(AC\) is the transversal: \(\angle BAC\) and \(\angle ACD\), \(\angle BCA\) and \(\angle CAD\).
• When \(AD\parallel BC\) and \(AC\) is the transversal: \(\angle DAC\) and \(\angle BCA\), \(\angle BAC\) and \(\angle ACD\).
• When \(AB\parallel CD\) and \(BD\) is the transversal: \(\angle ABD\) and \(\angle BDC\), \(\angle ADB\) and \(\angle DBC\).
• When \(AD\parallel BC\) and \(BD\) is the transversal: \(\angle ADB\) and \(\angle DBC\), \(\angle ABD\) and \(\angle BDC\).

The parallel lines forming each pair are \(AB\parallel CD\) and \(AD\parallel BC\).