Chapter 4.5, Problem 4WE

To determine

To prove: $\angle \text{TSA}\cong \angle \text{STR}$

Explanation of Solution

Given Information$\overline{\text{RT}}\cong \overline{\text{AS}}$ and $\overline{\text{RS}}\cong \overline{\text{AT}}$

Proof In $\Delta \text{TSR}$ and $\Delta \text{TSA}$

	Points	Reasons
(i)	$\overline{\text{RS}}\cong \overline{\text{AT}}$	Given
(ii)	$\overline{\text{RT}}\cong \overline{\text{AS}}$	Given
(iii)	$\overline{\text{ST}}=\overline{\text{TS}}$	Common in both triangles

So, by SSS congruency

$\Delta \text{TSR }\cong \Delta \text{TSA}$

Hence, $\angle \text{SRT}\cong \angle \text{TAS}$ ....(a)

By congruent part of congruent triangles,

$\angle \text{RST}\cong \angle \text{ATS}$...(b)

Now, In $\Delta \text{SOR}$ and $\Delta \text{TOA}$

	Points	Reasons
(i)	$\overline{\text{SR}}\cong \overline{\text{TA}}$	Given
(ii)	$\angle \text{SOR}\cong \angle \text{TOA}$	Vertically opposite angles
(iii)	$\angle \text{SRO}\cong \angle \text{TAO}$	Since, $\angle \text{SRT}\cong \angle \text{TAS}$ , from (a)

So, by AAS congruency

$\Delta \text{SOR }\cong \Delta \text{TOA}$

Hence, by congruent part of congruent triangles,

$\angle \text{RSO}\cong \angle \text{ATO}$...(c)

Now, from (b), we have

$\angle \text{RST}\cong \angle \text{ATS}$

So,

⇒ $\angle \text{RSO }+\angle \text{OST}\cong \angle \text{ATO}+\angle \text{OTS}$

⇒$\angle \text{TSO}\cong \angle \text{STO}$ [Since, $\angle \text{RSO}\cong \angle \text{ATO}$ , from (c)]

⇒ $\angle \text{TSA}\cong \angle \text{STR}$