notes - cpctc proofs 4.5 name antoine collins22. date 10/29/25
cpctc means
1)
given:
prove:
1.
2.
3.
4.
5.
6.
7.
2) given:
prove:
1.
2.
3.
4.
5.

Question

Accepted Answer

# Explanation:
## Step1: Reasons for Statements in First Proof
1. **Statement 1**: Given (as per problem statement)
2. **Statement 2**: If two parallel lines ($\overline{AB}\parallel\overline{CD}$) are cut by a transversal ($\overline{BD}$), then alternate - interior angles ($\angle ABD$ and $\angle CDB$) are congruent.
3. **Statement 3**: Given (as per problem statement)
4. **Statement 4**: If two parallel lines ($\overline{BC}\parallel\overline{AD}$) are cut by a transversal ($\overline{BD}$), then alternate - interior angles ($\angle CBD$ and $\angle ADB$) are congruent.
5. **Statement 5**: Reflexive property of congruence ($\overline{BD}\cong\overline{BD}$ because a segment is congruent to itself)
6. **Statement 6**: By the ASA (Angle - Side - Angle) congruence criterion ($\angle ABD\cong\angle CDB$, $\overline{BD}\cong\overline{BD}$, $\angle CBD\cong\angle ADB$), $\triangle ABD\cong\triangle CDB$
7. **Statement 7**: CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Since $\triangle ABD\cong\triangle CDB$, $\overline{AB}\cong\overline{CD}$

## Step2: Reasons for Statements in Second Proof
1. **Statement 1**: $\overline{QR}\cong\overline{QT}$ (Given)
2. **Statement 2**: $\overline{RS}\cong\overline{TS}$ (Given)
3. **Statement 3**: $\overline{QS}\cong\overline{QS}$ (Reflexive property of congruence)
4. **Statement 4**: By the SSS (Side - Side - Side) congruence criterion ($\overline{QR}\cong\overline{QT}$, $\overline{RS}\cong\overline{TS}$, $\overline{QS}\cong\overline{QS}$), $\triangle QRS\cong\triangle QTS$
5. **Statement 5**: CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Since $\triangle QRS\cong\triangle QTS$, $\angle SRQ\cong\angle STQ$

# Answer:
1.
| Statements | Reasons |
|---|---|
|1. $\overline{AB}\parallel\overline{CD}$| Given |
|2. $\angle ABD\cong\angle CDB$| Alternate - interior angles theorem |
|3. $\overline{BC}\parallel\overline{AD}$| Given |
|4. $\angle CBD\cong\angle ADB$| Alternate - interior angles theorem |
|5. $\overline{BD}\cong\overline{BD}$| Reflexive property |
|6. $\triangle ABD\cong\triangle CDB$| ASA |
|7. $\overline{AB}\cong\overline{CD}$| CPCTC |

2.
| Statements | Reasons |
|---|---|
|1. $\overline{QR}\cong\overline{QT}$| Given |
|2. $\overline{RS}\cong\overline{TS}$| Given |
|3. $\overline{QS}\cong\overline{QS}$| Reflexive property |
|4. $\triangle QRS\cong\triangle QTS$| SSS |
|5. $\angle SRQ\cong\angle STQ$| CPCTC |

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QUESTION IMAGE

Question

Explanation:

Step1: Reasons for Statements in First Proof

Step2: Reasons for Statements in Second Proof

Answer:

Statements	Reasons
2. \(\angle ABD\cong\angle CDB\)	Alternate - interior angles theorem
3. \(\overline{BC}\parallel\overline{AD}\)	Given
4. \(\angle CBD\cong\angle ADB\)	Alternate - interior angles theorem
5. \(\overline{BD}\cong\overline{BD}\)	Reflexive property
6. \(\triangle ABD\cong\triangle CDB\)	ASA
7. \(\overline{AB}\cong\overline{CD}\)	CPCTC

Statements	Reasons
2. \(\overline{RS}\cong\overline{TS}\)	Given
3. \(\overline{QS}\cong\overline{QS}\)	Reflexive property
4. \(\triangle QRS\cong\triangle QTS\)	SSS
5. \(\angle SRQ\cong\angle STQ\)	CPCTC