Geometry proof practice worksheet with ten problems requiring two-column proofs for triangle congruence using given conditions and diagrams.

Here are the step-by-step two-column proofs for each problem on the worksheet.

1) Prove: $\Delta ABC \cong \Delta DCB$
* Statement 1: $AB \parallel CD$; $AB \cong CD$ (Given)
* Statement 2: $\angle ABC \cong \angle DCB$ (Alternate Interior Angles Theorem)
* Statement 3: $BC \cong CB$ (Reflexive Property of Congruence)
* Statement 4: $\Delta ABC \cong \Delta DCB$ (SAS Postulate - Side Angle Side)

2) Prove: $\Delta JKN \cong \Delta MKL$
* Statement 1: $\angle J \cong \angle M$, K is midpoint of $\overline{JM}$ (Given)
* Statement 2: $\overline{JK} \cong \overline{MK}$ (Definition of Midpoint)
* Statement 3: $\angle JKN \cong \angle MKL$ (Vertical Angles Theorem)
* Statement 4: $\Delta JKN \cong \Delta MKL$ (ASA Postulate - Angle Side Angle)

3) Prove: $\Delta XTZ \cong \Delta WTZ$
* Statement 1: $\angle X \cong \angle W$, $\overline{ZT} \perp \overline{XW}$ (Given)
* Statement 2: $\angle ZTX \cong \angle ZTW$ (Definition of Perpendicular Lines creates Right Angles)
* Statement 3: $\overline{ZT} \cong \overline{ZT}$ (Reflexive Property of Congruence)
* Statement 4: $\Delta XTZ \cong \Delta WTZ$ (AAS Theorem - Angle Angle Side)

4) Prove: $\Delta RML \cong \Delta RMP$
* Statement 1: $\overline{RM} \perp \overline{LP}$, $\overline{RL} \cong \overline{RP}$ (Given)
* Statement 2: $\Delta RML$ and $\Delta RMP$ are right triangles (Definition of Perpendicular Lines)
* Statement 3: $\overline{RM} \cong \overline{RM}$ (Reflexive Property of Congruence)
* Statement 4: $\Delta RML \cong \Delta RMP$ (HL Theorem - Hypotenuse Leg)

5) Prove: $\Delta ABC \cong \Delta ADC$
* Statement 1: $\overline{AC}$ bisects $\angle BCD$, $\overline{BC} \cong \overline{DC}$ (Given)
* Statement 2: $\angle BCA \cong \angle DCA$ (Definition of Angle Bisector)
* Statement 3: $\overline{AC} \cong \overline{AC}$ (Reflexive Property of Congruence)
* Statement 4: $\Delta ABC \cong \Delta ADC$ (SAS Postulate - Side Angle Side)

6) Prove: $\Delta ABD \cong \Delta CDB$
* Statement 1: $\overline{AB} \cong \overline{DC}$, $\overline{AD} \cong \overline{BC}$ (Given)
* Statement 2: $\overline{BD} \cong \overline{DB}$ (Reflexive Property of Congruence)
* Statement 3: $\Delta ABD \cong \Delta CDB$ (SSS Postulate - Side Side Side)

7) Prove: $\Delta RST \cong \Delta UST$
* Statement 1: $\angle R \cong \angle U$, $\overline{ST}$ bisects $\angle RSU$ (Given)
* Statement 2: $\angle RST \cong \angle UST$ (Definition of Angle Bisector)
* Statement 3: $\overline{ST} \cong \overline{ST}$ (Reflexive Property of Congruence)
* Statement 4: $\Delta RST \cong \Delta UST$ (AAS Theorem - Angle Angle Side)

8) Prove: $\Delta BDA \cong \Delta BDC$
* Statement 1: $\angle BDA$ and $\angle BDC$ are right angles, $\overline{BA} \cong \overline{BC}$ (Given)
* Statement 2: $\Delta BDA$ and $\Delta BDC$ are right triangles (Definition of Right Angles)
* Statement 3: $\overline{BD} \cong \overline{BD}$ (Reflexive Property of Congruence)
* Statement 4: $\Delta BDA \cong \Delta BDC$ (HL Theorem - Hypotenuse Leg)

9) Prove: $\Delta RNM \cong \Delta RNP$
* Statement 1: $\overline{RN} \perp \overline{MP}$, N is midpoint of $\overline{MP}$ (Given)
* Statement 2: $\overline{MN} \cong \overline{PN}$ (Definition of Midpoint)
* Statement 3: $\angle RNM \cong \angle RNP$ (Perpendicular lines form congruent right angles)
* Statement 4: $\overline{RN} \cong \overline{RN}$ (Reflexive Property of Congruence)
* Statement 5: $\Delta RNM \cong \Delta RNP$ (SAS Postulate - Side Angle Side)

10) Prove: $\Delta ABC \cong \Delta ABD$
* Statement 1: $\angle C \cong \angle D$, $\overline{AB} \perp \overline{CD}$ (Given)
* Statement 2: $\angle ABC \cong \angle ABD$ (Perpendicular lines form congruent right angles)
* Statement 3: $\overline{AB} \cong \overline{AB}$ (Reflexive Property of Congruence)
* Statement 4: $\Delta ABC \cong \Delta ABD$ (AAS Theorem - Angle Angle Side)