Question

proving a theorem prove the alternate interior angles theorem.
statements
reasons
given
corresponding angles theorem
vertical angles congruence theorem
transitive property of angle congruence

Explanation:

Step1: Identify the given statement

The first blank under "STATEMENTS" with reason "Given" should be \(p\parallel q\) as it is the initial condition provided for proving the theorem.

Step2: Apply the Corresponding Angles Theorem

When \(p\parallel q\), by the Corresponding Angles Theorem, \(\angle1\cong\angle3\) (since \(\angle1\) and \(\angle3\) are corresponding angles formed by transversal \(t\) with parallel lines \(p\) and \(q\)).

Step3: Use the Vertical Angles Congruence Theorem

By the Vertical Angles Congruence Theorem, \(\angle1\cong\angle2\) (because \(\angle1\) and \(\angle2\) are vertical angles).

Step4: Apply the Transitive Property of Angle Congruence

Since \(\angle1\cong\angle3\) (from Step 2) and \(\angle1\cong\angle2\) (from Step 3), by the Transitive Property of Angle Congruence, \(\angle3\cong\angle2\).

Answer:

First blank (Given): \(p\parallel q\)
Second blank (Corresponding Angles Theorem): \(\angle1\cong\angle3\)
Third blank (Vertical Angles Congruence Theorem): \(\angle1\cong\angle2\)
Fourth blank (Transitive Property of Angle Congruence): \(\angle3\cong\angle2\)