Question

In Exercises 1–3 you are given a diagram that is marked with given information. Give the reason for each key step of the proof.

Prove:$\overline{\mathrm{AS}}\cong \overline{\mathrm{DT}}$ Key steps of proof:

a.$\mathrm{\Delta }\mathbf{ABC}\cong \mathrm{\Delta }\mathbf{DEF}$

b.$\angle \mathbf{C}\cong \angle \mathbf{F}$

c.$\mathrm{\Delta }\mathbf{ACS}\cong \mathrm{\Delta }\mathbf{DFT}$

d.$\overline{\mathrm{AS}}\cong \overline{\mathrm{DT}}$

Short answer

a.$\mathrm{\Delta }ABC\cong \mathrm{\Delta }DEF$	SAS postulate
b.$\angle C\cong \angle F$	Corr. parts of$\cong \mathrm{\Delta }$ are $\cong$.
c.$\mathrm{\Delta }ACS\cong \mathrm{\Delta }DFT$	SAS postulate
d.$\overline{AS}\cong \overline{DT}$	Corr. parts of$\cong \mathrm{\Delta }$ are $\cong$.

Step-by-step solution

Step 1. Apply SAS postulate.

Consider$\mathrm{\Delta }ABC$ and$\mathrm{\Delta }DEF$ in which $\overline{AB}\cong \overline{DE}$,$\angle BAC\cong \angle EDF$ and$\overline{AC}\cong \overline{DF}$ then by SAS postulate $\mathrm{\Delta }ABC\cong \mathrm{\Delta }DEF$.

Step 2. Description of step.

As $\mathrm{\Delta }ABC\cong \mathrm{\Delta }DEF$, by corresponding parts of congruent triangles, $\angle C\cong \angle F$.

Step 3. Apply SAS postulate.

Consider $\mathrm{\Delta }ACS$and $\mathrm{\Delta }DFT$ in which $\overline{AC}\cong \overline{DF}$,$\angle C\cong \angle F$ and$\overline{SC}\cong \overline{TF}$ then by SAS postulate $\mathrm{\Delta }ACS\cong \mathrm{\Delta }DFT$.

Step 4. Description of step.

As $\mathrm{\Delta }ACS\cong \mathrm{\Delta }DFT$, by corresponding parts of congruent triangles, $\overline{AS}\cong \overline{DT}$.

Step 5. Give the reason for each key step of the proof.

a.$\mathrm{\Delta }ABC\cong \mathrm{\Delta }DEF$	SAS postulate
b.$\angle C\cong \angle F$	Corr. parts of role="math" localid="1649948016680" $\cong \mathrm{\Delta }$are $\cong$.
c.$\mathrm{\Delta }ACS\cong \mathrm{\Delta }DFT$	SAS postulate
d.$\overline{AS}\cong \overline{DT}$	Corr. parts of$\cong \mathrm{\Delta }$ are $\cong$.