name: jayda mancey unit 4: congruent triangles
date: nov 19, 25 bell: homework 6: proving triangles congruent:
asa, aas, and hl.
** this is a 2 - page document! **
complete proofs 1 - 2 using angle - side - angle (asa).
1. given: $ \overline{bd} $ bisects $ \angle abc, \angle bda\cong\angle bdc $
prove: $ \triangle abd\cong\triangle cbd $
2. given: $ \overline{st}\parallel\overline{uv}, \overline{st}\cong\overline{uv} $
prove: $ \triangle stw\cong\triangle vuw $
complete proofs 3 - 5 using angle - angle - side (aas).
3. given: $ \overline{fh} $ bisects $ \angle ehg, \angle feh\cong\angle fgh $
prove: $ \triangle feh\cong\triangle fgh $

Question

Accepted Answer

# Explanation:
## Step1: Analyze the first proof (ASA - $\triangle ABD\cong\triangle CBD$)
- **Statement 3**: $BD = BD$ (common side, reflexive property of equality).
- **Statement 4**: Since $BD$ bisects $\angle ABC$, $\angle ABD=\angle CBD$ (definition of an angle - bisector).
- **Reason 5**: By the Angle - Side - Angle (ASA) congruence criterion ($\angle ABD=\angle CBD$, $BD = BD$, $\angle BDA=\angle BDC$), $\triangle ABD\cong\triangle CBD$.

## Step2: Analyze the second proof (ASA - $\triangle STW\cong\triangle VUW$)
- **Statement 3**: $\angle TSW\cong\angle UVW$ (corresponding angles for parallel lines $ST\parallel UV$ and transversal $SWV$).
- **Reason 5**: By the Angle - Side - Angle (ASA) congruence criterion ($\angle TSW\cong\angle UVW$, $ST = UV$, $\angle TWS=\angle UWV$ (vertical angles are congruent)), $\triangle STW\cong\triangle VUW$.

## Step3: Analyze the third proof (AAS - $\triangle FEH\cong\triangle FGH$)
- **Statement 4**: Since $FH$ bisects $\angle EHG$, $\angle EHF=\angle GHF$ (definition of an angle - bisector).
- **Statement 5**: $FH = FH$ (common side, reflexive property of equality).
- **Reason 5**: By the Angle - Angle - Side (AAS) congruence criterion ($\angle FEH=\angle FGH$, $\angle EHF=\angle GHF$, $FH = FH$), $\triangle FEH\cong\triangle FGH$.

# Answer:
1.
| Statements | Reasons |
|---|---|
| 1. $BD$ bisects $\angle ABC$ | 1. Given |
| 2. $\angle BDA\cong\angle BDC$ | 2. Given |
| 3. $BD = BD$ | 3. Reflexive property |
| 4. $\angle ABD=\angle CBD$ | 4. Definition of angle - bisector |
| 5. $\triangle ABD\cong\triangle CBD$ | 5. ASA |

2.
| Statements | Reasons |
|---|---|
| 1. $ST\parallel UV$ | 1. Given |
| 2. $ST\cong UV$ | 2. Given |
| 3. $\angle TSW\cong\angle UVW$ | 3. Corresponding angles (parallel lines) |
| 4. $\angle TWS=\angle UWV$ | 4. Vertical angles |
| 5. $\triangle STW\cong\triangle VUW$ | 5. ASA |

3.
| Statements | Reasons |
|---|---|
| 1. $FH$ bisects $\angle EHG$ | 1. Given |
| 2. $\angle FEH\cong\angle FGH$ | 2. Given |
| 3. $FH = FH$ | 3. Reflexive property |
| 4. $\angle EHF=\angle GHF$ | 4. Definition of angle - bisector |
| 5. $\triangle FEH\cong\triangle FGH$ | 5. AAS |

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Analyze the first proof (ASA - \(\triangle ABD\cong\triangle CBD\))

Step2: Analyze the second proof (ASA - \(\triangle STW\cong\triangle VUW\))

Step3: Analyze the third proof (AAS - \(\triangle FEH\cong\triangle FGH\))

Answer:

Statements	Reasons
2. \(\angle BDA\cong\angle BDC\)	2. Given
3. \(BD = BD\)	3. Reflexive property
4. \(\angle ABD=\angle CBD\)	4. Definition of angle - bisector
5. \(\triangle ABD\cong\triangle CBD\)	5. ASA

Statements	Reasons
2. \(ST\cong UV\)	2. Given
3. \(\angle TSW\cong\angle UVW\)	3. Corresponding angles (parallel lines)
4. \(\angle TWS=\angle UWV\)	4. Vertical angles
5. \(\triangle STW\cong\triangle VUW\)	5. ASA

Statements	Reasons
2. \(\angle FEH\cong\angle FGH\)	2. Given
3. \(FH = FH\)	3. Reflexive property
4. \(\angle EHF=\angle GHF\)	4. Definition of angle - bisector
5. \(\triangle FEH\cong\triangle FGH\)	5. AAS