Question

complete the proofs for the following.
23)
given:
prove:
\\( \overline { a b } \parallel \overline { d c } \\)
\\( \angle b \cong \angle d \\)
\\( \triangle a d c \cong \triangle c b a \\)
24)
given:
prove:
\\( \overline { a b } \cong \overline { c b } \\)
\\( \overline { a d } \cong \overline { c d } \\)
\\( \triangle a b d \cong \triangle c b d \\)

Explanation:

Problem 23:

Step-by-Step Proof:

We are given \( \overline{AB} \parallel \overline{DC} \) and \( \angle B \cong \angle D \), and we need to prove \( \triangle ADC \cong \triangle CBA \).

Step 1: Identify the given information

• \( \angle B \cong \angle D \) (Given)
• \( \overline{AB} \parallel \overline{DC} \) (Given)

Step 2: Use the Alternate Interior Angles Theorem

Since \( \overline{AB} \parallel \overline{DC} \) and \( \overline{AC} \) is a transversal, we have \( \angle BAC \cong \angle DCA \) (Alternate Interior Angles Theorem).

Step 3: Identify the common side

\( \overline{AC} \cong \overline{AC} \) (Reflexive Property of Congruence).

Step 4: Apply the AAS (Angle-Angle-Side) Congruence Criterion

We have:

• \( \angle D \cong \angle B \) (Given)
• \( \angle DCA \cong \angle BAC \) (Alternate Interior Angles)
• \( \overline{AC} \cong \overline{AC} \) (Reflexive Property)

By the AAS Congruence Criterion, \( \triangle ADC \cong \triangle CBA \).

Problem 24:

Step-by-Step Proof:

We are given \( \overline{AB} \cong \overline{CB} \) and \( \overline{AD} \cong \overline{CD} \), and we need to prove \( \triangle ABD \cong \triangle CBD \).

Step 1: Identify the given information

• \( \overline{AB} \cong \overline{CB} \) (Given)
• \( \overline{AD} \cong \overline{CD} \) (Given)

Step 2: Identify the common side

\( \overline{BD} \cong \overline{BD} \) (Reflexive Property of Congruence).

Step 3: Apply the SSS (Side-Side-Side) Congruence Criterion

We have:

• \( \overline{AB} \cong \overline{CB} \) (Given)
• \( \overline{AD} \cong \overline{CD} \) (Given)
• \( \overline{BD} \cong \overline{BD} \) (Reflexive Property)

By the SSS Congruence Criterion, \( \triangle ABD \cong \triangle CBD \).

Final Answers:

• Problem 23: \( \triangle ADC \cong \triangle CBA \) by AAS.
• Problem 24: \( \triangle ABD \cong \triangle CBD \) by SSS.

Answer:

Problem 23:

Step-by-Step Proof:

We are given \( \overline{AB} \parallel \overline{DC} \) and \( \angle B \cong \angle D \), and we need to prove \( \triangle ADC \cong \triangle CBA \).

Step 1: Identify the given information

• \( \angle B \cong \angle D \) (Given)
• \( \overline{AB} \parallel \overline{DC} \) (Given)

Step 2: Use the Alternate Interior Angles Theorem

Since \( \overline{AB} \parallel \overline{DC} \) and \( \overline{AC} \) is a transversal, we have \( \angle BAC \cong \angle DCA \) (Alternate Interior Angles Theorem).

Step 3: Identify the common side

\( \overline{AC} \cong \overline{AC} \) (Reflexive Property of Congruence).

Step 4: Apply the AAS (Angle-Angle-Side) Congruence Criterion

We have:

• \( \angle D \cong \angle B \) (Given)
• \( \angle DCA \cong \angle BAC \) (Alternate Interior Angles)
• \( \overline{AC} \cong \overline{AC} \) (Reflexive Property)

By the AAS Congruence Criterion, \( \triangle ADC \cong \triangle CBA \).

Problem 24:

Step-by-Step Proof:

We are given \( \overline{AB} \cong \overline{CB} \) and \( \overline{AD} \cong \overline{CD} \), and we need to prove \( \triangle ABD \cong \triangle CBD \).

Step 1: Identify the given information

• \( \overline{AB} \cong \overline{CB} \) (Given)
• \( \overline{AD} \cong \overline{CD} \) (Given)

Step 2: Identify the common side

\( \overline{BD} \cong \overline{BD} \) (Reflexive Property of Congruence).

Step 3: Apply the SSS (Side-Side-Side) Congruence Criterion

We have:

• \( \overline{AB} \cong \overline{CB} \) (Given)
• \( \overline{AD} \cong \overline{CD} \) (Given)
• \( \overline{BD} \cong \overline{BD} \) (Reflexive Property)

By the SSS Congruence Criterion, \( \triangle ABD \cong \triangle CBD \).

Final Answers:

• Problem 23: \( \triangle ADC \cong \triangle CBA \) by AAS.
• Problem 24: \( \triangle ABD \cong \triangle CBD \) by SSS.