Question

tu ≅ qv, st ≅ qr, and uv ≅ rs. complete the proof that δstu ≅ δrqv.

Explanation:

Step1: Identify given congruences

Given: $\overline{TU} \cong \overline{QV}$, $\overline{ST} \cong \overline{QR}$, $\overline{UV} \cong \overline{RS}$

Step2: Segment addition postulate for $\overline{RV}$

$\overline{RV} = \overline{UV} + \overline{RU}$ (Segment Addition Postulate)

Step3: Segment addition postulate for $\overline{SU}$

$\overline{SU} = \overline{RS} + \overline{RU}$ (Segment Addition Postulate)

Step4: Substitute congruent segments

Since $\overline{UV} \cong \overline{RS}$, substitute $\overline{UV}$ with $\overline{RS}$: $\overline{RV} = \overline{RS} + \overline{RU}$ (Substitution Property of Equality)

Step5: Transitive property for segments

$\overline{SU} = \overline{RV}$ (Transitive Property of Equality, from Steps 3 and 4)

Step6: SSS congruence criterion

$\triangle STU \cong \triangle RQV$ (SSS Congruence: $\overline{ST} \cong \overline{QR}$, $\overline{TU} \cong \overline{QV}$, $\overline{SU} \cong \overline{RV}$)

Answer:

Statement	Reason
$\overline{ST} \cong \overline{QR}$	Given
$\overline{UV} \cong \overline{RS}$	Given
$\overline{RV} = \overline{UV} + \overline{RU}$	Segment Addition Postulate
$\overline{SU} = \overline{RS} + \overline{RU}$	Segment Addition Postulate
$\overline{RV} = \overline{RS} + \overline{RU}$	Substitution Property of Equality
$\overline{SU} = \overline{RV}$	Transitive Property of Equality
$\triangle STU \cong \triangle RQV$	SSS Triangle Congruence Postulate