Geometry proof practice worksheet with ten problems requiring two-column proofs for triangle congruence.

Since the user has set both `ExplainAnswer` and `TeachTopic` to false, I will proceed directly to solving the problems presented in the image. The task is to write a two-column proof for each of the 10 geometry problems shown.

I will go through each problem one by one, identifying the given information, the goal, and the logical steps (theorems/postulates) needed to bridge the gap between them. Then I will format these into standard two-column proofs.

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Problem 1
* Given: $\overline{AB} \parallel \overline{CD}$; $\overline{AB} \cong \overline{CD}$
* Prove: $\Delta ABC \cong \Delta DCB$
* Reasoning:
1. We have parallel lines $\overline{AB}$ and $\overline{CD}$ cut by transversal $\overline{BC}$. This creates alternate interior angles that are congruent: $\angle ABC \cong \angle DCB$.
2. We are given side $\overline{AB} \cong \overline{CD}$.
3. Both triangles share side $\overline{BC}$. By the Reflexive Property, $\overline{BC} \cong \overline{CB}$.
4. We now have Side-Angle-Side (SAS): Side ($\overline{AB}$), Angle ($\angle ABC$), Side ($\overline{BC}$).

Problem 2
* Given: $\angle J \cong \angle M$, $K$ is midpoint of $\overline{JM}$
* Prove: $\Delta JKN \cong \Delta MKL$
* Reasoning:
1. Since $K$ is the midpoint of $\overline{JM}$, segment $\overline{JK} \cong \overline{MK}$.
2. Angles $\angle JKN$ and $\angle MKL$ are vertical angles, so they are congruent.
3. We are given $\angle J \cong \angle M$.
4. We have Angle-Side-Angle (ASA): Angle ($\angle J$), Side ($\overline{JK}$), Angle ($\angle JKN$).

Problem 3
* Given: $\angle X \cong \angle W$, $\overline{ZT} \perp \overline{XW}$
* Prove: $\Delta XTZ \cong \Delta WTZ$
* Reasoning:
1. Perpendicular lines create right angles. So $\angle ZTX$ and $\angle ZTW$ are right angles and thus congruent.
2. We are given $\angle X \cong \angle W$.
3. Both triangles share side $\overline{ZT}$. By Reflexive Property, $\overline{ZT} \cong \overline{ZT}$.
4. We have Angle-Angle-Side (AAS): Angle ($\angle X$), Angle ($\angle ZTX$), Side ($\overline{ZT}$). Note: While AAS works, we could also deduce the third angle is congruent and use ASA, but AAS is direct here.

Problem 4
* Given: $\overline{RM} \perp \overline{LP}$; $\overline{RL} \cong \overline{RP}$
* Prove: $\Delta RML \cong \Delta RMP$
* Reasoning:
1. Perpendicular lines create right angles at $M$. So $\Delta RML$ and $\Delta RMP$ are right triangles.
2. Hypotenuses $\overline{RL}$ and $\overline{RP}$ are given as congruent.
3. Leg $\overline{RM}$ is shared by both triangles (Reflexive Property).
4. This fits the Hypotenuse-Leg (HL) theorem for right triangles.

Problem 5
* Given: $\overline{AC}$ bisects $\angle BCD$; $\overline{BC} \cong \overline{DC}$
* Prove: $\Delta ABC \cong \Delta ADC$
* Reasoning:
1. An angle bisector divides an angle into two congruent angles. So $\angle BCA \cong \angle DCA$.
2. We are given side $\overline{BC} \cong \overline{DC}$.
3. Both triangles share side $\overline{AC}$ (Reflexive Property).
4. We have Side-Angle-Side (SAS): Side ($\overline{BC}$), Angle ($\angle BCA$), Side ($\overline{AC}$).

Problem 6
* Given: $\overline{AB} \cong \overline{DC}$; $\overline{AD} \cong \overline{BC}$
* Prove: $\Delta ABD \cong \Delta CDB$
* Reasoning:
1. We are given two pairs of congruent sides: $\overline{AB} \cong \overline{DC}$ and $\overline{AD} \cong \overline{BC}$.
2. Both triangles share diagonal $\overline{BD}$. By Reflexive Property, $\overline{BD} \cong \overline{DB}$.
3. We have three pairs of congruent sides: Side-Side-Side (SSS).

Problem 7
* Given: $\angle R \cong \angle U$; $\overline{ST}$ bisects $\angle RSU$
* Prove: $\Delta RST \cong \Delta UST$
* Reasoning:
1. Angle bisector $\overline{ST}$ means $\angle RST \cong \angle UST$.
2. We are given $\angle R \cong \angle U$.
3. Side $\overline{ST}$ is shared (Reflexive Property).
4. We have Angle-Angle-Side (AAS): Angle ($\angle R$), Angle ($\angle RST$), Side ($\overline{ST}$).

Problem 8
* Given: $\angle BDA$ and $\angle BDC$ are right angles; $\overline{BA} \cong \overline{BC}$
* Prove: $\Delta BDA \cong \Delta BDC$
* Reasoning:
1. Right angles mean these are right triangles.
2. Hypotenuses $\overline{BA}$ and $\overline{BC}$ are given congruent.
3. Leg $\overline{BD}$ is shared (Reflexive Property).
4. Use Hypotenuse-Leg (HL) theorem.

Problem 9
* Given: $\overline{RN} \perp \overline{MP}$; $N$ is midpoint of $\overline{MP}$
* Prove: $\Delta RNM \cong \Delta RNP$
* Reasoning:
1. Midpoint $N$ means $\overline{MN} \cong \overline{PN}$.
2. Perpendicular $\overline{RN} \perp \overline{MP}$ means $\angle RNM$ and $\angle RNP$ are right angles and therefore congruent.
3. Side $\overline{RN}$ is shared (Reflexive Property).
4. We have Side-Angle-Side (SAS): Side ($\overline{MN}$), Angle ($\angle RNM$), Side ($\overline{RN}$).

Problem 10
* Given: $\angle C \cong \angle D$; $\overline{AB} \perp \overline{CD}$
* Prove: $\Delta ABC \cong \Delta ABD$
* Reasoning:
1. Perpendicular $\overline{AB} \perp \overline{CD}$ means $\angle ABC$ and $\angle ABD$ are right angles and congruent.
2. We are given $\angle C \cong \angle D$.
3. Side $\overline{AB}$ is shared (Reflexive Property).
4. We have Angle-Angle-Side (AAS): Angle ($\angle C$), Angle ($\angle ABC$), Side ($\overline{AB}$).

All logic checks out. I will now present the final answer containing all 10 proofs.

Final Answer:
Here are the two-column proofs for each problem.

Problem 1

| Statements | Reasons |
| :--- | :--- |
| 1. $\overline{AB} \parallel \overline{CD}$; $\overline{AB} \cong \overline{CD}$ | 1. Given |
| 2. $\angle ABC \cong \angle DCB$ | 2. Alternate Interior Angles Theorem |
| 3. $\overline{BC} \cong \overline{CB}$ | 3. Reflexive Property of Congruence |
| 4. $\Delta ABC \cong \Delta DCB$ | 4. SAS Postulate |

Problem 2

| Statements | Reasons |
| :--- | :--- |
| 1. $\angle J \cong \angle M$; $K$ is midpoint of $\overline{JM}$ | 1. Given |
| 2. $\overline{JK} \cong \overline{MK}$ | 2. Definition of Midpoint |
| 3. $\angle JKN \cong \angle MKL$ | 3. Vertical Angles Theorem |
| 4. $\Delta JKN \cong \Delta MKL$ | 4. ASA Postulate |

Problem 3

| Statements | Reasons |
| :--- | :--- |
| 1. $\angle X \cong \angle W$; $\overline{ZT} \perp \overline{XW}$ | 1. Given |
| 2. $\angle ZTX \cong \angle ZTW$ | 2. Definition of Perpendicular Lines |
| 3. $\overline{ZT} \cong \overline{ZT}$ | 3. Reflexive Property of Congruence |
| 4. $\Delta XTZ \cong \Delta WTZ$ | 4. AAS Theorem |

Problem 4

| Statements | Reasons |
| :--- | :--- |
| 1. $\overline{RM} \perp \overline{LP}$; $\overline{RL} \cong \overline{RP}$ | 1. Given |
| 2. $\Delta RML$ and $\Delta RMP$ are Rt. $\Delta$s | 2. Definition of Perpendicular Lines |
| 3. $\overline{RM} \cong \overline{RM}$ | 3. Reflexive Property of Congruence |
| 4. $\Delta RML \cong \Delta RMP$ | 4. HL Theorem |

Problem 5

| Statements | Reasons |
| :--- | :--- |
| 1. $\overline{AC}$ bisects $\angle BCD$; $\overline{BC} \cong \overline{DC}$ | 1. Given |
| 2. $\angle BCA \cong \angle DCA$ | 2. Definition of Angle Bisector |
| 3. $\overline{AC} \cong \overline{AC}$ | 3. Reflexive Property of Congruence |
| 4. $\Delta ABC \cong \Delta ADC$ | 4. SAS Postulate |

Problem 6

| Statements | Reasons |
| :--- | :--- |
| 1. $\overline{AB} \cong \overline{DC}$; $\overline{AD} \cong \overline{BC}$ | 1. Given |
| 2. $\overline{BD} \cong \overline{DB}$ | 2. Reflexive Property of Congruence |
| 3. $\Delta ABD \cong \Delta CDB$ | 3. SSS Postulate |

Problem 7

| Statements | Reasons |
| :--- | :--- |
| 1. $\angle R \cong \angle U$; $\overline{ST}$ bisects $\angle RSU$ | 1. Given |
| 2. $\angle RST \cong \angle UST$ | 2. Definition of Angle Bisector |
| 3. $\overline{ST} \cong \overline{ST}$ | 3. Reflexive Property of Congruence |
| 4. $\Delta RST \cong \Delta UST$ | 4. AAS Theorem |

Problem 8

| Statements | Reasons |
| :--- | :--- |
| 1. $\angle BDA$ and $\angle BDC$ are right angles; $\overline{BA} \cong \overline{BC}$ | 1. Given |
| 2. $\Delta BDA$ and $\Delta BDC$ are Rt. $\Delta$s | 2. Definition of Right Triangles |
| 3. $\overline{BD} \cong \overline{BD}$ | 3. Reflexive Property of Congruence |
| 4. $\Delta BDA \cong \Delta BDC$ | 4. HL Theorem |

Problem 9

| Statements | Reasons |
| :--- | :--- |
| 1. $\overline{RN} \perp \overline{MP}$; $N$ is midpoint of $\overline{MP}$ | 1. Given |
| 2. $\angle RNM \cong \angle RNP$ | 2. Definition of Perpendicular Lines |
| 3. $\overline{MN} \cong \overline{PN}$ | 3. Definition of Midpoint |
| 4. $\overline{RN} \cong \overline{RN}$ | 4. Reflexive Property of Congruence |
| 5. $\Delta RNM \cong \Delta RNP$ | 5. SAS Postulate |

Problem 10

| Statements | Reasons |
| :--- | :--- |
| 1. $\angle C \cong \angle D$; $\overline{AB} \perp \overline{CD}$ | 1. Given |
| 2. $\angle ABC \cong \angle ABD$ | 2. Definition of Perpendicular Lines |
| 3. $\overline{AB} \cong \overline{AB}$ | 3. Reflexive Property of Congruence |
| 4. $\Delta ABC \cong \Delta ABD$ | 4. AAS Theorem |