Question

given \\(\overline{pq}\\) bisects \\(\angle spt\\), \\(\overline{sp} \cong \overline{tp}\\)\
prove \\(\triangle spq \cong \triangle tpq\\)\
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statements\
\\(\overline{sp} \cong \overline{tp}\\), \\(\overline{pq}\\) bisects \\(\angle spt\\)\
\\(\overline{pq} \cong \overline{pq}\\)\
\\(\triangle spq \cong \triangle tpq\\)\
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reasons\
given\
blank\
definition of angle bisector\
blank

Explanation:

Step1: Reason for $\overline{PQ}\cong\overline{PQ}$

Reflexive property of congruence (a segment is congruent to itself).

Step2: Fill the statement for angle - bisector

Since $\overline{PQ}$ bisects $\angle SPT$, by the definition of angle - bisector, $\angle SPQ\cong\angle TPQ$.

Step3: Prove $\triangle SPQ\cong\triangle TPQ$

We have $\overline{SP}\cong\overline{TP}$ (given), $\angle SPQ\cong\angle TPQ$ (from angle - bisector definition), and $\overline{PQ}\cong\overline{PQ}$ (reflexive property). By the Side - Angle - Side (SAS) congruence criterion, $\triangle SPQ\cong\triangle TPQ$.

Answer:

• For $\overline{PQ}\cong\overline{PQ}$, the reason is "Reflexive property of congruence".
• The statement for the angle - bisector is "$\angle SPQ\cong\angle TPQ$".
• For $\triangle SPQ\cong\triangle TPQ$, the reason is "Side - Angle - Side (SAS) congruence criterion".