Geometry Files: Triangle Congruence - Insert Clever Math Pun Here - Free Printable

Problem 1:

Given: $\overline{AB} \parallel \overline{CD}$, $\overline{AB} \cong \overline{CD}$
Prove: $\triangle ABC \cong \triangle DCB$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\overline{AB} \parallel \overline{CD}$ | Given |
| 2. $\overline{AB} \cong \overline{CD}$ | Given |
| 3. $\angle ABC \cong \angle DCB$ | Alternate Interior Angles Theorem (since $\overline{AB} \parallel \overline{CD}$ and $\overline{BC}$ is a transversal) |
| 4. $\overline{BC} \cong \overline{CB}$ | Reflexive Property of Congruence |
| 5. $\triangle ABC \cong \triangle DCB$ | Side-Angle-Side (SAS) Congruence Postulate |

---

Problem 2:

Given: $\angle J \cong \angle M$, $K$ is the midpoint of $\overline{JM}$
Prove: $\triangle JKN \cong \triangle MKL$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\angle J \cong \angle M$ | Given |
| 2. $K$ is the midpoint of $\overline{JM}$ | Given |
| 3. $\overline{JK} \cong \overline{MK}$ | Definition of Midpoint |
| 4. $\angle JKN \cong \angle MKL$ | Vertical Angles Theorem |
| 5. $\triangle JKN \cong \triangle MKL$ | Angle-Side-Angle (ASA) Congruence Postulate |

---

Problem 3:

Given: $\angle X \cong \angle W$, $\overline{ZT} \perp \overline{XW}$
Prove: $\triangle XTZ \cong \triangle WTZ$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\angle X \cong \angle W$ | Given |
| 2. $\overline{ZT} \perp \overline{XW}$ | Given |
| 3. $\angle ZTX \cong \angle ZTW$ | Definition of Perpendicular Lines (both are right angles) |
| 4. $\overline{ZT} \cong \overline{ZT}$ | Reflexive Property of Congruence |
| 5. $\triangle XTZ \cong \triangle WTZ$ | Angle-Angle-Side (AAS) Congruence Theorem |

---

Problem 4:

Given: $\overline{RM} \perp \overline{LP}$, $\overline{RL} \cong \overline{RP}$
Prove: $\triangle RML \cong \triangle RMP$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\overline{RM} \perp \overline{LP}$ | Given |
| 2. $\angle RML \cong \angle RMP$ | Definition of Perpendicular Lines (both are right angles) |
| 3. $\overline{RL} \cong \overline{RP}$ | Given |
| 4. $\overline{RM} \cong \overline{RM}$ | Reflexive Property of Congruence |
| 5. $\triangle RML \cong \triangle RMP$ | Hypotenuse-Leg (HL) Congruence Theorem |

---

Problem 5:

Given: $\overline{AC}$ bisects $\angle BCD$, $\overline{BC} \cong \overline{DC}$
Prove: $\triangle ABC \cong \triangle ADC$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\overline{AC}$ bisects $\angle BCD$ | Given |
| 2. $\angle BCA \cong \angle DCA$ | Definition of Angle Bisector |
| 3. $\overline{BC} \cong \overline{DC}$ | Given |
| 4. $\overline{AC} \cong \overline{AC}$ | Reflexive Property of Congruence |
| 5. $\triangle ABC \cong \triangle ADC$ | Side-Angle-Side (SAS) Congruence Postulate |

---

Problem 6:

Given: $\overline{AB} \cong \overline{DC}$, $\overline{AD} \cong \overline{BC}$
Prove: $\triangle ABD \cong \triangle CDB$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\overline{AB} \cong \overline{DC}$ | Given |
| 2. $\overline{AD} \cong \overline{BC}$ | Given |
| 3. $\overline{BD} \cong \overline{DB}$ | Reflexive Property of Congruence |
| 4. $\triangle ABD \cong \triangle CDB$ | Side-Side-Side (SSS) Congruence Postulate |

---

Problem 7:

Given: $\angle R \cong \angle U$, $\overline{ST}$ bisects $\angle RSU$
Prove: $\triangle RST \cong \triangle UST$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\angle R \cong \angle U$ | Given |
| 2. $\overline{ST}$ bisects $\angle RSU$ | Given |
| 3. $\angle RST \cong \angle UST$ | Definition of Angle Bisector |
| 4. $\overline{ST} \cong \overline{ST}$ | Reflexive Property of Congruence |
| 5. $\triangle RST \cong \triangle UST$ | Angle-Side-Angle (ASA) Congruence Postulate |

---

Problem 8:

Given: $\angle BDA$ and $\angle BDC$ are right angles, $\overline{BA} \cong \overline{BC}$
Prove: $\triangle BDA \cong \triangle BDC$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\angle BDA$ and $\angle BDC$ are right angles | Given |
| 2. $\overline{BA} \cong \overline{BC}$ | Given |
| 3. $\overline{BD} \cong \overline{BD}$ | Reflexive Property of Congruence |
| 4. $\triangle BDA \cong \triangle BDC$ | Hypotenuse-Leg (HL) Congruence Theorem |

---

Problem 9:

Given: $\overline{RN} \perp \overline{MP}$, $N$ is the midpoint of $\overline{MP}$
Prove: $\triangle RNM \cong \triangle RNP$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\overline{RN} \perp \overline{MP}$ | Given |
| 2. $\angle RNM \cong \angle RNP$ | Definition of Perpendicular Lines (both are right angles) |
| 3. $N$ is the midpoint of $\overline{MP}$ | Given |
| 4. $\overline{MN} \cong \overline{PN}$ | Definition of Midpoint |
| 5. $\overline{RN} \cong \overline{RN}$ | Reflexive Property of Congruence |
| 6. $\triangle RNM \cong \triangle RNP$ | Leg-Leg (LL) Congruence Theorem for Right Triangles |

---

Problem 10:

Given: $\angle C \cong \angle D$, $\overline{AB} \perp \overline{CD}$
Prove: $\triangle ABC \cong \triangle ABD$

#### Proof:
| Statements | Reasons |
|-----------------|--------------|
| 1. $\angle C \cong \angle D$ | Given |
| 2. $\overline{AB} \perp \overline{CD}$ | Given |
| 3. $\angle ABC \cong \angle ABD$ | Definition of Perpendicular Lines (both are right angles) |
| 4. $\overline{AB} \cong \overline{AB}$ | Reflexive Property of Congruence |
| 5. $\triangle ABC \cong \triangle ABD$ | Angle-Angle-Side (AAS) Congruence Theorem |

---

Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Problem 1: } \triangle ABC \cong \triangle DCB \text{ (SAS)} \\
\text{Problem 2: } \triangle JKN \cong \triangle MKL \text{ (ASA)} \\
\text{Problem 3: } \triangle XTZ \cong \triangle WTZ \text{ (AAS)} \\
\text{Problem 4: } \triangle RML \cong \triangle RMP \text{ (HL)} \\
\text{Problem 5: } \triangle ABC \cong \triangle ADC \text{ (SAS)} \\
\text{Problem 6: } \triangle ABD \cong \triangle CDB \text{ (SSS)} \\
\text{Problem 7: } \triangle RST \cong \triangle UST \text{ (ASA)} \\
\text{Problem 8: } \triangle BDA \cong \triangle BDC \text{ (HL)} \\
\text{Problem 9: } \triangle RNM \cong \triangle RNP \text{ (LL)} \\
\text{Problem 10: } \triangle ABC \cong \triangle ABD \text{ (AAS)}
\end{array}
}
\]