Question

question 24 of 45
what is the missing statement for the sixth step in the proof below?
given: t is the mid - point of \\(\overline{qr}\\)
u is the mid - point of \\(\overline{rs}\\)
prove: \\(\triangle qrs\sim\triangle tru\\)
statements\treasons

1. t is the mid - point of \\(\overline{qr}\\)

u is the mid - point of \\(\overline{rs}\\)\tgiven

1. \\(\overline{qt}\cong\overline{tr}\\), \\(\overline{ru}\cong\overline{us}\\), \\(\overline{ru}+\overline{us}=\overline{rs}\\)\tdefinition of a mid - point
2. \\(\overline{qr}=\overline{qt}+\overline{tr}=\overline{tr}+\overline{tr}=2\overline{tr}\\)

\\(\overline{rs}=\overline{ru}+\overline{us}=\overline{ru}+\overline{ru}=2\overline{ru}\\)\tsubstitution and simplification

1. \\(\frac{\overline{qr}}{\overline{rs}}=\frac{2\overline{tr}}{2\overline{ru}}=\frac{\overline{tr}}{\overline{ru}}\\)\tdivision property of equality, simplification
2. \\(\frac{\overline{qr}}{\overline{rs}}=\frac{\overline{rt}}{\overline{ru}}\\)\tproperty of proportion from interchangeability
3. ?\treflexive property
4. \\(\triangle qrs\sim\triangle tru\\)\tsas similarity postulate

a. \\(\overline{qs}\cong\overline{qs}\\)
b. \\(\angle qrs\cong\angle tru\\)
c. \\(\overline{ru}\cong\overline{us}\\)
d. \\(\angle tqs\cong\angle rqs\\)

Explanation:

The reflexive property is often used for congruence of a segment to itself or an angle to itself. In the context of proving triangle similarity using SAS (Side - Angle - Side), we need an angle congruence. Since we are dealing with two triangles $\triangle QRS$ and $\triangle TRU$, and we already have side - ratio equalities in previous steps, we need an included - angle congruence. $\angle QRS$ and $\angle TRU$ are the same angle (reflexive property for angles), which is the included angle between the sides in the proportion.

Answer:

B. $\angle QRS\cong\angle TRU$