Question

complete the proof that ( mangle rqt = mangle p + mangle t ) without using the triangle angle sum theorem.

(image of a geometric figure with points p, q, t, r, s and a table with statements and reasons)

1. ( overline{qs} parallel overline{pt} ) - given
2. ( angle p cong angle rqs ) -
3. ( angle t cong angle sqt ) -
4. ( mangle rqt = mangle rqs + mangle sqt ) - additive property of angle measure
5. ( mangle rqt = mangle p + mangle t ) -

Explanation:

Step1: Analyze ∠P ≅ ∠RQS

Since \(\overline{QS} \parallel \overline{PT}\), \( \angle P \) and \( \angle RQS \) are corresponding angles. By the Corresponding Angles Postulate (when two parallel lines are cut by a transversal, corresponding angles are congruent), \( \angle P \cong \angle RQS \).

Step2: Analyze ∠T ≅ ∠SQT

Again, because \(\overline{QS} \parallel \overline{PT}\), \( \angle T \) and \( \angle SQT \) are alternate interior angles. By the Alternate Interior Angles Theorem (when two parallel lines are cut by a transversal, alternate interior angles are congruent), \( \angle T \cong \angle SQT \).

Step3: Analyze the last step

We know \( m\angle RQT = m\angle RQS + m\angle SQT \) (Additive Property of Angle Measure) and from steps 2 and 3, \( m\angle RQS = m\angle P \) and \( m\angle SQT = m\angle T \). By the Substitution Property (replacing \( m\angle RQS \) with \( m\angle P \) and \( m\angle SQT \) with \( m\angle T \) in the equation), we get \( m\angle RQT = m\angle P + m\angle T \).

Filling the Reasons:

• For Statement 2: Corresponding Angles Postulate (since \( \overline{QS} \parallel \overline{PT} \) and \( \overline{QP} \) is a transversal, \( \angle P \) and \( \angle RQS \) are corresponding angles).
• For Statement 3: Alternate Interior Angles Theorem (since \( \overline{QS} \parallel \overline{PT} \) and \( \overline{QT} \) is a transversal, \( \angle T \) and \( \angle SQT \) are alternate interior angles).
• For Statement 5: Substitution Property (substituting \( m\angle RQS \) with \( m\angle P \) and \( m\angle SQT \) with \( m\angle T \) from Statements 2 and 3 into Statement 4).

Answer:

• Statement 2 Reason: Corresponding Angles Postulate
• Statement 3 Reason: Alternate Interior Angles Theorem
• Statement 5 Reason: Substitution Property