Question

$overline{uv} cong overline{tu}$ and $overline{vw} cong overline{tw}$. complete the proof that $overline{vx} cong overline{tx}$.

(image of a geometric figure with points u, t, v, w, x and a table with statements and reasons for a proof)

	statement	reason
1	$overline{uv} cong overline{tu}$	given
2	$overline{vw} cong overline{tw}$	given
3	$overline{uw} cong overline{uw}$	reflexive property of congruence
4	$\triangle tuw cong \triangle vuw$	sss
5	$angle tuw cong angle vuw$	cpctc
6	$overline{ux} cong overline{ux}$	reflexive property of congruence
7	$\triangle tux cong \triangle vux$	sas
8	$overline{vx} cong overline{tx}$	blank

Explanation:

Step1: Recall CPCTC

CPCTC (Corresponding Parts of Congruent Triangles are Congruent) states that if two triangles are congruent, their corresponding parts (sides and angles) are congruent.

Step2: Apply CPCTC to $\triangle TUX$ and $\triangle VUX$

We know from step 7 that $\triangle TUX \cong \triangle VUX$ (by SAS). So, the corresponding sides $\overline{VX}$ and $\overline{TX}$ must be congruent by CPCTC.

Answer:

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)