Question

In Exercises 1-6 you are given a diagram that is marked with given information. Give the reason for each key step of the proof.

Prove:$\overline{\mathrm{DX}}\cong \overline{\mathrm{EX}}$

Key steps of proof:

a.$\mathrm{\Delta }\mathbf{POD}\cong \mathrm{\Delta }\mathbf{POE}$

b.$\overline{\mathrm{PD}}\cong \overline{\mathrm{PE}}$

c.$\mathrm{\Delta }\mathbf{PDX}\cong \mathrm{\Delta }\mathbf{PEX}$

d.$\overline{\mathrm{DX}}\cong \overline{\mathrm{EX}}$

Short answer

a.$\mathrm{\Delta }POD\cong \mathrm{\Delta }POE$	SAS postulate
b.$\overline{PD}\cong \overline{PE}$	Corr. parts of$\cong \mathrm{\Delta }$ are $\cong$.
c.$\mathrm{\Delta }PDX\cong \mathrm{\Delta }PEX$	SAS postulate
d.$\overline{DX}\cong \overline{EX}$	Corr. parts of$\cong \mathrm{\Delta }$ are $\cong$.

Step-by-step solution

Step 1. Apply SAS postulate.

Consider$\mathrm{\Delta }POD$ and$\mathrm{\Delta }POE$ in which$\overline{OD}\cong \overline{OE},\overline{PO}\cong \overline{PO}$ and$\angle POD\cong \angle POE$ then by SAS postulate $\mathrm{\Delta }POD\cong \mathrm{\Delta }POE$.

Step 2. Description of step.

As $\mathrm{\Delta }POD\cong \mathrm{\Delta }POE$, by corresponding parts of congruent triangles, $\overline{PD}\cong \overline{PE}$.

Step 3. Apply SAS postulate.

Consider $\mathrm{\Delta }PDX$and $\mathrm{\Delta }PEX$ in which$\overline{PD}\cong \overline{PE},\overline{PX}\cong \overline{PX}$ and$\angle PXD\cong \angle PXE$ then by SAS postulate $\mathrm{\Delta }PDX\cong \mathrm{\Delta }PEX$.

Step 4. Description of step.

As $\mathrm{\Delta }PDX\cong \mathrm{\Delta }PEX$, by corresponding parts of congruent triangles, $\overline{DX}\cong \overline{EX}$.

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

a.$\mathrm{\Delta }POD\cong \mathrm{\Delta }POE$	SAS postulate
b.$\overline{PD}\cong \overline{PE}$	Corr. parts of$\cong \mathrm{\Delta }$are $\cong$.
c.$\mathrm{\Delta }PDX\cong \mathrm{\Delta }PEX$	SAS postulate
d.$\overline{DX}\cong \overline{EX}$	Corr. parts of$\cong \mathrm{\Delta }$ are $\cong$.