Question

$\overline{tv} \cong \overline{su}$, $\overline{sw} \parallel \overline{tv}$, and $\overline{st} \parallel \overline{uv}$. complete the proof that $\angle tuv \cong \angle svu$.

	statement	reason
2	$\overline{sw} \parallel \overline{tv}$	given
3	$\overline{st} \parallel \overline{uv}$	given
4	$\overline{st} \parallel \overline{vw}$	given
5	$\overline{tv} \cong \overline{sw}$	parallelograms have congruent opposite sides
6	$\overline{sw} \cong \overline{su}$
7	$\angle suw \cong \angle w$
8	$\angle tvu \cong \angle w$	corresponding angles theorem
9	$\angle tvu \cong \angle suw$	transitive property of congruence
10	$\overline{uv} \cong \overline{uv}$	reflexive property of congruence
11	$\triangle suv \cong \triangle tvu$	sas
12	$\angle tuv \cong \angle svu$	cpctc

Explanation:

For Statement 6:

Step1: Identify known congruences

We know from Statement 1 that $\overline{TV} \cong \overline{SU}$ and from Statement 5 that $\overline{TV} \cong \overline{SW}$.

Step2: Apply Transitive Property

The Transitive Property of Congruence states that if $a \cong b$ and $b \cong c$, then $a \cong c$. Here, let $a = \overline{SW}$, $b = \overline{TV}$, and $c = \overline{SU}$. Since $\overline{TV} \cong \overline{SU}$ (Statement 1) and $\overline{TV} \cong \overline{SW}$ (Statement 5), by the Transitive Property of Congruence, we can conclude that $\overline{SW} \cong \overline{SU}$.

For Statement 7:

Step1: Identify triangle type

Since $\overline{SW} \cong \overline{SU}$ (from Statement 6), triangle $SUW$ is isosceles with $\overline{SW} = \overline{SU}$.

Step2: Apply Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. In $\triangle SUW$, the sides $\overline{SW}$ and $\overline{SU}$ are congruent, so the angles opposite them, $\angle SUW$ and $\angle W$, are congruent. So the reason is "Isosceles Triangle Theorem" (or "Base Angles of an Isosceles Triangle are Congruent").

Final Answers for the Blanks:

• Statement 6 Reason: Transitive Property of Congruence
• Statement 7 Reason: Isosceles Triangle Theorem (or Base Angles of an Isosceles Triangle are Congruent)

(If we assume we need to fill one at a time, for Statement 6: Transitive Property of Congruence; for Statement 7: Isosceles Triangle Theorem)

Answer:

Step1: Identify triangle type

Since $\overline{SW} \cong \overline{SU}$ (from Statement 6), triangle $SUW$ is isosceles with $\overline{SW} = \overline{SU}$.

Step2: Apply Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. In $\triangle SUW$, the sides $\overline{SW}$ and $\overline{SU}$ are congruent, so the angles opposite them, $\angle SUW$ and $\angle W$, are congruent. So the reason is "Isosceles Triangle Theorem" (or "Base Angles of an Isosceles Triangle are Congruent").

Final Answers for the Blanks:

• Statement 6 Reason: Transitive Property of Congruence
• Statement 7 Reason: Isosceles Triangle Theorem (or Base Angles of an Isosceles Triangle are Congruent)

(If we assume we need to fill one at a time, for Statement 6: Transitive Property of Congruence; for Statement 7: Isosceles Triangle Theorem)