Question

problem 24
given: parallelogram spqt,
\\( \overline { p k } \cong \overline { t l } \\)
prove: \\( \angle p s k \cong \angle t q l \\)

1. \\( \overline { p k } \cong \overline { t l } \\)
2. given

6.
6.

1. \\( \angle p s k \cong \angle t q l \\)

7.

Explanation:

Step1: Properties of parallelogram

In parallelogram \(SPQT\), \(SP\parallel QT\) and \(SP = QT\). So, \(\angle SPK=\angle TQL\) (alternate - interior angles) and \(\angle S\) \(PK=\angle TQL\) (alternate - interior angles).

Step2: Use SAS (Side - Angle - Side) congruence

We know that \(PK\cong TL\) (given), \(\angle SPK=\angle TQL\) (from step 1) and \(SP = QT\) (property of parallelogram). By SAS congruence criterion, \(\triangle SPK\cong\triangle TQL\).

Step3: Corresponding parts of congruent triangles

Since \(\triangle SPK\cong\triangle TQL\), then \(\angle PSK\cong\angle TQL\) (corresponding parts of congruent triangles are congruent).

Answer:

1. \(\triangle SPK\cong\triangle TQL\)
2. SAS (Side - Angle - Side) congruence criterion (\(SP = QT\) (property of parallelogram), \(\angle SPK=\angle TQL\) (alternate - interior angles), \(PK\cong TL\) (given))
3. Corresponding parts of congruent triangles are congruent.