Question

statements reasons 1. ? 1. given 2. ? 2. definition of perpendicular. 3. ? 3. all right angles are congruent. 4. ? 4. def. of segment bisector 5. ? 5. reflexive property of ≅ 6. ? 6. sas ∠bda ≅ ∠bdc bd ≅ bd ad ≅ cd ∠bda and ∠bdc are rt. angles. bd ⊥ ac, bd bisects ac. △abd ≅ △cbd

Explanation:

To solve the proof for triangle congruence (\(\triangle ABD \cong \triangle CBD\)), we analyze each step:

Step 1: Identify the Given Statement

The first statement in a proof is typically the "Given" information. From the diagram and context, the given is:
\(\boldsymbol{BD \perp AC}\) (BD is perpendicular to AC) and \(\boldsymbol{BD}\) bisects \(AC\) (so \(AD = CD\)).

Step 2: Definition of Perpendicular

If \(BD \perp AC\), by the definition of perpendicular, \(\angle BDA\) and \(\angle BDC\) are right angles (since perpendicular lines form \(90^\circ\) angles). Thus:
\(\boldsymbol{\angle BDA \text{ and } \angle BDC \text{ are right angles}}\).

Step 3: All Right Angles are Congruent

By the property that "all right angles are congruent," we conclude:
\(\boldsymbol{\angle BDA \cong \angle BDC}\).

Step 4: Reflexive Property of Congruence

A segment is congruent to itself (reflexive property). Thus:
\(\boldsymbol{BD \cong BD}\).

Step 5: Definition of Segment Bisector

Since \(BD\) bisects \(AC\), the segment bisector definition tells us \(AD = CD\), so:
\(\boldsymbol{AD \cong CD}\).

Step 6: SAS Congruence Criterion

To prove \(\triangle ABD \cong \triangle CBD\), we use the SAS (Side-Angle-Side) criterion:

• Side: \(AD \cong CD\) (from Step 5),
• Angle: \(\angle BDA \cong \angle BDC\) (from Step 3),
• Side: \(BD \cong BD\) (from Step 4).

Thus, by SAS:
\(\boldsymbol{\triangle ABD \cong \triangle CBD}\).

Final Proof Table (Statements → Reasons)

Statements	Reasons
2. \(\angle BDA\), \(\angle BDC\) are right angles	2. Definition of perpendicular
3. \(\angle BDA \cong \angle BDC\)	3. All right angles are congruent
4. \(BD \cong BD\)	4. Reflexive property of \(\cong\)
5. \(AD \cong CD\)	5. Definition of segment bisector
6. \(\triangle ABD \cong \triangle CBD\)	6. SAS (Side-Angle-Side)

This completes the proof using the SAS congruence criterion.

Answer:

To solve the proof for triangle congruence (\(\triangle ABD \cong \triangle CBD\)), we analyze each step:

Step 1: Identify the Given Statement

The first statement in a proof is typically the "Given" information. From the diagram and context, the given is:
\(\boldsymbol{BD \perp AC}\) (BD is perpendicular to AC) and \(\boldsymbol{BD}\) bisects \(AC\) (so \(AD = CD\)).

Step 2: Definition of Perpendicular

If \(BD \perp AC\), by the definition of perpendicular, \(\angle BDA\) and \(\angle BDC\) are right angles (since perpendicular lines form \(90^\circ\) angles). Thus:
\(\boldsymbol{\angle BDA \text{ and } \angle BDC \text{ are right angles}}\).

Step 3: All Right Angles are Congruent

By the property that "all right angles are congruent," we conclude:
\(\boldsymbol{\angle BDA \cong \angle BDC}\).

Step 4: Reflexive Property of Congruence

A segment is congruent to itself (reflexive property). Thus:
\(\boldsymbol{BD \cong BD}\).

Step 5: Definition of Segment Bisector

Since \(BD\) bisects \(AC\), the segment bisector definition tells us \(AD = CD\), so:
\(\boldsymbol{AD \cong CD}\).

Step 6: SAS Congruence Criterion

To prove \(\triangle ABD \cong \triangle CBD\), we use the SAS (Side-Angle-Side) criterion:

• Side: \(AD \cong CD\) (from Step 5),
• Angle: \(\angle BDA \cong \angle BDC\) (from Step 3),
• Side: \(BD \cong BD\) (from Step 4).

Thus, by SAS:
\(\boldsymbol{\triangle ABD \cong \triangle CBD}\).

Final Proof Table (Statements → Reasons)

Statements	Reasons
2. \(\angle BDA\), \(\angle BDC\) are right angles	2. Definition of perpendicular
3. \(\angle BDA \cong \angle BDC\)	3. All right angles are congruent
4. \(BD \cong BD\)	4. Reflexive property of \(\cong\)
5. \(AD \cong CD\)	5. Definition of segment bisector
6. \(\triangle ABD \cong \triangle CBD\)	6. SAS (Side-Angle-Side)

This completes the proof using the SAS congruence criterion.