Geometry proofs practice worksheet with ten problems involving triangle congruence, parallel lines, midpoints, and right angles.

Final Answer:
Problem 1:
| Statement | Reason |
|-----------|--------|
| $\overline{AB} \parallel \overline{CD}$ | Given |
| $\angle ABC \cong \angle DCB$ | Alternate interior angles (parallel lines cut by transversal $\overline{BC}$) |
| $\overline{AB} \cong \overline{CD}$ | Given |
| $\overline{BC} \cong \overline{CB}$ | Reflexive property |
| $\triangle ABC \cong \triangle DCB$ | SAS Congruence |

Problem 2:
| Statement | Reason |
|-----------|--------|
| $\angle J \cong \angle M$ | Given |
| $K$ is midpoint of $\overline{JM}$ | Given |
| $\overline{JK} \cong \overline{MK}$ | Definition of midpoint |
| $\angle JKL \cong \angle MKL$ | Vertical angles are congruent |
| $\triangle JKN \cong \triangle MKL$ | ASA Congruence |

Problem 3:
| Statement | Reason |
|-----------|--------|
| $\angle X \cong \angle W$ | Given |
| $\overline{ZT} \perp \overline{XW}$ | Given |
| $\angle XTZ \cong \angle WTZ$ | Both are right angles (definition of perpendicular) |
| $\overline{ZT} \cong \overline{ZT}$ | Reflexive property |
| $\triangle XTZ \cong \triangle WTZ$ | AAS Congruence |

Problem 4:
| Statement | Reason |
|-----------|--------|
| $\overline{RM} \perp \overline{LP}$ | Given |
| $\angle RML \cong \angle RMP$ | Both are right angles |
| $\overline{RL} \cong \overline{RP}$ | Given |
| $\overline{RM} \cong \overline{RM}$ | Reflexive property |
| $\triangle RML \cong \triangle RMP$ | HL Congruence (right triangles) |

Problem 5:
| Statement | Reason |
|-----------|--------|
| $\overline{AC}$ bisects $\angle BCD$ | Given |
| $\angle BCA \cong \angle DCA$ | Definition of angle bisector |
| $\overline{BC} \cong \overline{DC}$ | Given |
| $\overline{AC} \cong \overline{AC}$ | Reflexive property |
| $\triangle ABC \cong \triangle ADC$ | SAS Congruence |

Problem 6:
| Statement | Reason |
|-----------|--------|
| $\overline{AB} \cong \overline{DC}$ | Given |
| $\overline{AD} \cong \overline{BC}$ | Given |
| $\overline{BD} \cong \overline{DB}$ | Reflexive property |
| $\triangle ABD \cong \triangle CDB$ | SSS Congruence |

Problem 7:
| Statement | Reason |
|-----------|--------|
| $\angle R \cong \angle U$ | Given |
| $\overline{ST}$ bisects $\angle RSU$ | Given |
| $\angle RST \cong \angle UST$ | Definition of angle bisector |
| $\overline{ST} \cong \overline{ST}$ | Reflexive property |
| $\triangle RST \cong \triangle UST$ | AAS Congruence |

Problem 8:
| Statement | Reason |
|-----------|--------|
| $\angle BDA$ and $\angle BDC$ are right angles | Given |
| $\angle BDA \cong \angle BDC$ | All right angles are congruent |
| $\overline{BA} \cong \overline{BC}$ | Given |
| $\overline{BD} \cong \overline{BD}$ | Reflexive property |
| $\triangle BDA \cong \triangle BDC$ | HL Congruence |

Problem 9:
| Statement | Reason |
|-----------|--------|
| $\overline{RN} \perp \overline{MP}$ | Given |
| $\angle RNM \cong \angle RNP$ | Both are right angles |
| $N$ is midpoint of $\overline{MP}$ | Given |
| $\overline{MN} \cong \overline{PN}$ | Definition of midpoint |
| $\overline{RN} \cong \overline{RN}$ | Reflexive property |
| $\triangle RNM \cong \triangle RNP$ | SAS Congruence |

Problem 10:
| Statement | Reason |
|-----------|--------|
| $\angle C \cong \angle D$ | Given |
| $\overline{AB} \perp \overline{CD}$ | Given |
| $\angle ABC \cong \angle ABD$ | Both are right angles |
| $\overline{AB} \cong \overline{AB}$ | Reflexive property |
| $\triangle ABC \cong \triangle ABD$ | AAS Congruence |