Question

complete the proof that ( mangle q + mangle u + mangle qru = 180^circ ) without using the triangle angle sum theorem.

statement	reason
2 ( angle q cong angle srt )
3 ( angle u cong angle tru )
4 ( mangle sru = mangle srt + mangle tru )
5 ( mangle sru = mangle q + mangle u )
6 ( mangle sru + mangle qru = 180^circ )
7 ( mangle q + mangle u + mangle qru = 180^circ )

Explanation:

Step1: Identify Parallel Lines Reason

From the diagram, \( \overline{QU} \parallel \overline{RT} \) is given (as per the problem's diagram and setup, so the reason is "Given").

Step2: Corresponding Angles Reason

\( \angle Q \cong \angle SRT \) because they are corresponding angles (since \( \overline{QU} \parallel \overline{RT} \) and \( \overline{SR} \) is a transversal), so the reason is "Corresponding Angles Postulate".

Step3: Alternate Interior Angles Reason

\( \angle U \cong \angle TRU \) because they are alternate interior angles (since \( \overline{QU} \parallel \overline{RT} \) and \( \overline{RU} \) is a transversal), so the reason is "Alternate Interior Angles Theorem".

Step4: Angle Addition Postulate Reason

\( m\angle SRU = m\angle SRT + m\angle TRU \) by the Angle Addition Postulate (which states that if a point lies in the interior of an angle, the sum of the two smaller angles equals the larger angle).

Step5: Substitution Property Reason

\( m\angle SRU = m\angle Q + m\angle U \) by substituting \( \angle Q \cong \angle SRT \) and \( \angle U \cong \angle TRU \) (from steps 2 and 3), so the reason is "Substitution Property of Equality".

Step6: Linear Pair Postulate Reason

\( m\angle SRU + m\angle QRU = 180^\circ \) because \( \angle SRU \) and \( \angle QRU \) form a linear pair (they are adjacent and form a straight line), so the reason is "Linear Pair Postulate" (linear pairs are supplementary, summing to \( 180^\circ \)).

Step7: Substitution Property (Final) Reason

\( m\angle Q + m\angle U + m\angle QRU = 180^\circ \) by substituting \( m\angle SRU = m\angle Q + m\angle U \) into \( m\angle SRU + m\angle QRU = 180^\circ \), so the reason is "Substitution Property of Equality".

Answer:

1. Given
2. Corresponding Angles Postulate
3. Alternate Interior Angles Theorem
4. Angle Addition Postulate
5. Substitution Property of Equality
6. Linear Pair Postulate (supplementary angles)
7. Substitution Property of Equality