Step-by-step explanation:
It is given that ∠SPQ≅∠SRQ
.
The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Angles S P Q and S R Q are highlighted in red.
It is also given that SQ⎯⎯⎯⎯⎯
bisects ∠PSR
.
By the definition of angle bisector, ∠PSQ≅∠QSR
.
The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Angles P S Q and R S Q are congruent and highlighted in red.
△PQS
and △RQS share a common side SQ⎯⎯⎯⎯⎯, and SQ⎯⎯⎯⎯⎯≅SQ⎯⎯⎯⎯⎯
by the Reflexive Property of Congruence.
The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Segment S Q is highlighted in red.
Two angles, ∠SPQ
and ∠PSQ, and a nonincluded side, SQ⎯⎯⎯⎯⎯, of △PQS are congruent to two angles, ∠SRQ and ∠QSR, and a nonincluded side, SQ⎯⎯⎯⎯⎯, of △RQS
.
The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Angles P S Q and R S Q are highlighted in red. Angles S P Q and S R Q are highlighted in red. Side S Q is highlighted in blue.
So, △PQS≅△RQS
by the Angle-Angle-Side (AAS) Congruence Theorem.
PS⎯⎯⎯⎯⎯
and SR⎯⎯⎯⎯⎯ are corresponding sides of congruent triangles, △PQS and △RQS. So, PS⎯⎯⎯⎯⎯≅SR⎯⎯⎯⎯⎯
by CPCTC.
The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Sides S R and S P are congruent and highlighted in red.
Translate these six statements and reasons into a 2
-column proof,
1. ∠SPQ≅∠SRQ
(Given)
2. SQ⎯⎯⎯⎯⎯
bisects ∠PSR
. (Given)
3. ∠PSQ≅∠QSR
(Def. of ∠
bisect)
4. SQ⎯⎯⎯⎯⎯≅SQ⎯⎯⎯⎯⎯
(Reflex. Prop. of ≅
)
5. △PQS≅△RQS
(AAS Steps 1, 3, 4)
6. PS⎯⎯⎯⎯⎯≅SR⎯⎯⎯⎯⎯
(CPCTC)
There you go
.
is such that 
.
.
, we have:
.
.