options for ♣:
definition of adjacent angles
definition of angle bisector
definition of congruence
vertical angles are congruent

options for ♦:
aas
asa
sas
sss

The proof with the statement is given below.

Step-by-step explanation:

Given:

Construction of angle bisector i.e SP is the bisector of angle RPQ.

To prove:

Δ PWS ≅ Δ PXS

Proof:

In  Δ PWS  and Δ PXS  

∠ WPS ≅ ∠ XPS    …………..{definition of angle bisector}

SP ≅ SP                …….{Reflexive property}

PW ≅ XP              ……….{definition of ≅}

Δ PWS  ≅ Δ PXS  …{Side-Angle-Side test i .e SAS}

Angle bisector: A ray divides angle into two equal measures then the ray is called as angle bisector of the bisected angle.

In the construction SP is the bisector of the angle ∠ RPQ

SAS: This test is to prove the triangle congruent when two sides are congruent and angle between that sides should be congruent. then we can say the triangle is congruent by side angle side test.

is there supposed to be a picture here?