Question

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1. ( overline{p x} cong overline{p x} ) reflexive property of congruence

6 sas congruence postulate

1. corresponding parts of congruent triangles are congruent
2. point ( p ) is equidistant from the endpoints of ( overline{a b} ). definition of equidistant

( overline{p a} cong overline{p b} quad angle p x a cong angle p x b quad \triangle a x p cong \triangle a x q quad \triangle a x p cong \triangle b x p )
( angle p a x cong angle p x b )

Explanation:

Step1: Use SAS Congruence Postulate

For two triangles \(\triangle AXP\) and \(\triangle BXP\), if we assume \(AX = BX\) (from previous steps not shown here, but common in such set - ups), \(\angle AXP=\angle BXP\) (another given or derived angle equality), and \(PX = PX\) (from step 5, reflexive property). Then by the SAS (Side - Angle - Side) Congruence Postulate, \(\triangle AXP\cong\triangle BXP\).

Step2: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle AXP\cong\triangle BXP\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, the corresponding sides \(\overline{PA}\) and \(\overline{PB}\) are congruent, i.e., \(\overline{PA}\cong\overline{PB}\).

Answer:

1. \(\triangle AXP\cong\triangle BXP\)
2. \(\overline{PA}\cong\overline{PB}\)