Question

1. given \\(\angle ptq = \angle rqs\\), prove \\(\delta pqt\\) is isosceles.

Explanation:

Identify the given information and target

We are given the angle equality \(\angle PTQ = \angle RQS\). Our goal is to prove that \(\Delta PQT\) is an isosceles triangle. To do this, we can show that two of its angles are equal, specifically \(\angle PTQ\) and \(\angle PQT\).

Analyze vertical angles at intersection Q

Using the Vertical Angles knowledge point

$$ \angle PQT = \angle RQS $$

Apply transitivity to relate angles in triangle PQT

Using the Angle Congruence Transitivity knowledge point

$$
LATEXBLOCK0
$$

Conclude triangle classification

Using the Isosceles Triangle Theorem knowledge point

$$
LATEXBLOCK1
$$

Answer:

Here is the complete two-column proof:

Statement	Reason
2. \(\angle PQT = \angle RQS\)	2. Vertical Angles Theorem
3. \(\angle PTQ = \angle PQT\)	3. Transitive Property of Congruence
4. \(PQ = PT\)	4. Converse of the Isosceles Triangle Theorem (If two angles of a triangle are congruent, the sides opposite them are congruent)
5. \(\Delta PQT\) is isosceles	5. Definition of an isosceles triangle (a triangle with at least two congruent sides)