Question

$\overline{qs} \parallel \overline{pt}$. complete the proof that $m\angle p + m\angle t + m\angle pqt = 180\degree$ without using the triangle angle sum theorem.

	statement	reason
2	$\angle p \cong \angle rqs$
3	$\angle t \cong \angle sqt$
4	$m\angle rqt = m\angle rqs + m\angle sqt$
5	$m\angle rqt = m\angle p + m\angle t$
6	$m\angle rqt + m\angle pqt = 180\degree$
7	$m\angle p + m\angle t + m\angle pqt = 180\degree$

Explanation:

Step1: Identify Given

$\overline{QS} \parallel \overline{PT}$ (Given)

Step2: Corresponding Angles

$\angle P \cong \angle RQS$ (Corresponding Angles Postulate, since $QS \parallel PT$ and $PR$ is transversal)

Step3: Alternate Interior Angles

$\angle T \cong \angle SQT$ (Alternate Interior Angles Theorem, since $QS \parallel PT$ and $QT$ is transversal)

Step4: Angle Addition Postulate

$m\angle RQT = m\angle RQS + m\angle SQT$ (Angle Addition Postulate)

Step5: Substitution Property

$m\angle RQT = m\angle P + m\angle T$ (Substitution, from steps 2 and 3)

Step6: Linear Pair Postulate

$m\angle RQT + m\angle PQT = 180^\circ$ (Linear Pair Postulate, since $\angle RQT$ and $\angle PQT$ form a linear pair)

Step7: Substitution Property

$m\angle P + m\angle T + m\angle PQT = 180^\circ$ (Substitution, from steps 5 and 6)

Answer:

The proof is completed using corresponding angles, alternate interior angles, angle addition, linear pair postulate, and substitution properties as shown in the steps. The final conclusion is $m\angle P + m\angle T + m\angle PQT = 180^\circ$.