Question

△qrt is equilateral, t is the midpoint of \\(\overline{su}\\), and \\(\overline{rs} \cong \overline{qu}\\). complete the proof that \\(\triangle rst \cong \triangle qut\\).

image of a geometric figure with points s, t, u, r, q and a table with statement and reason columns:

1. \\(\triangle qrt\\) is equilateral
2. t is the midpoint of \\(\overline{su}\\)
3. \\(\overline{rs} \cong \overline{qu}\\)
4. \\(\overline{tu} \cong \overline{st}\\)
5. \\(\overline{qt} \cong \overline{rt}\\)
6. \\(\triangle rst \cong \triangle qut\\)

Explanation:

Step1: Analyze Statement 1

The problem states $\triangle QRT$ is equilateral. By the definition of an equilateral triangle, all its sides are equal. So the reason for statement 1 is "Given" (since it's provided in the problem).

Step2: Analyze Statement 2

$T$ is the midpoint of $\overline{SU}$. By the definition of a midpoint, a midpoint divides a segment into two congruent segments. So the reason for statement 2 is "Given" (as it's part of the problem's information).

Step3: Analyze Statement 3

$\overline{RS} \cong \overline{QU}$ is given in the problem, so the reason is "Given".

Step4: Analyze Statement 4

Since $T$ is the midpoint of $\overline{SU}$, by the definition of midpoint, $\overline{TU} \cong \overline{ST}$. So the reason is "Definition of midpoint".

Step5: Analyze Statement 5

$\triangle QRT$ is equilateral, so all sides are equal, which means $\overline{QT} \cong \overline{RT}$ (by definition of equilateral triangle: all sides of an equilateral triangle are congruent). So the reason is "All sides of an equilateral triangle are congruent".

Step6: Analyze Statement 6

We have $\overline{RS} \cong \overline{QU}$ (statement 3), $\overline{ST} \cong \overline{TU}$ (statement 4), and $\overline{RT} \cong \overline{QT}$ (statement 5). By the Side - Side - Side (SSS) Congruence Postulate, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. So $\triangle RST \cong \triangle QUT$ by SSS.

Answer:

1. Reason for $\triangle QRT$ is equilateral: Given
2. Reason for $T$ is the midpoint of $\overline{SU}$: Given
3. Reason for $\overline{RS} \cong \overline{QU}$: Given
4. Reason for $\overline{TU} \cong \overline{ST}$: Definition of midpoint
5. Reason for $\overline{QT} \cong \overline{RT}$: All sides of an equilateral triangle are congruent
6. Reason for $\triangle RST \cong \triangle QUT$: SSS Congruence Postulate