Question

complete the proof that $\triangle qsu \cong \triangle qtr$.

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

1. $\angle tqu \cong \angle rqs$ (given)
2. $\angle qsu \cong \angle qtr$ (given)
3. $\overline{qt} \cong \overline{qs}$ (given)
4. $m\angle squ = m\angle sqt + m\angle tqu$ (additive property of angle measure)
5. $m\angle rqt = m\angle rqs + m\angle sqt$ (additive property of angle measure)
6. $m\angle squ = m\angle sqt + m\angle rqs$ (reason to be filled)
7. $m\angle rqt = m\angle squ$ (transitive property of equality)
8. $\triangle qsu \cong \triangle qtr$ (reason to be filled))

Explanation:

Step1: Analyze Step 6 Reason

From Step 1, we know \( \angle TQU \cong \angle RQS \), so their measures are equal (\( m\angle TQU = m\angle RQS \)). In Step 4, \( m\angle SQU = m\angle SQT + m\angle TQU \). Substitute \( m\angle TQU \) with \( m\angle RQS \) (from Step 1, using Substitution Property of Equality) to get \( m\angle SQU = m\angle SQT + m\angle RQS \). So the reason for Step 6 is Substitution Property of Equality (since we substitute \( m\angle TQU \) with \( m\angle RQS \) as they are equal from Step 1).

Step2: Analyze Step 8 Reason

We have:

• \( \angle QSU \cong \angle QTR \) (Step 2, given)
• \( \overline{QT} \cong \overline{QS} \) (Step 3, given)
• \( \angle RQT \cong \angle SQU \) (from Step 7, since \( m\angle RQT = m\angle SQU \) implies the angles are congruent)

So by the Angle - Angle - Side (AAS) Congruence Criterion (two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle), \( \triangle QSU \cong \triangle QTR \).

Answer:

• Step 6 Reason: Substitution Property of Equality
• Step 8 Reason: AAS (Angle - Angle - Side) Congruence Criterion