Question

complete the proof by the following
23)
given:
prove:
24)
given:
prove:

Explanation:

To solve the problem of proving the congruence of triangles, we will use the Side - Side - Side (SSS) or Side - Angle - Side (SAS) congruence criteria. Let's analyze the two given triangle - related problems one by one.

Problem 1 (First Triangle Diagram)

Step 1: Identify the given information

From the diagram and the given data, we know that \(AB\parallel DC\) (given), \(AB = DC\) (given), and \(AC\) is a common side. Also, since \(AB\parallel DC\), \(\angle BAC=\angle DCA\) (alternate interior angles).

Step 2: Choose the congruence criterion

We have two sides and the included angle equal. \(AB = DC\), \(\angle BAC=\angle DCA\), and \(AC = AC\). So, by the SAS (Side - Angle - Side) congruence criterion, \(\triangle ABC\cong\triangle CDA\).

Problem 2 (Second Triangle Diagram: Prove \(\triangle ABD\cong\triangle CBD\))

Step 1: Identify the given information

We are given that \(AB = CB\), \(AD = CD\), and \(BD\) is a common side.

Step 2: Choose the congruence criterion

We have three sides equal: \(AB = CB\), \(AD = CD\), and \(BD=BD\). So, by the SSS (Side - Side - Side) congruence criterion, \(\triangle ABD\cong\triangle CBD\).

For the first proof ( \(\triangle ABC\cong\triangle CDA\)):

Statement	Reason
2. \(\angle BAC=\angle DCA\)	Alternate interior angles (since \(AB\parallel DC\))
3. \(AB = DC\)	Given
4. \(AC = AC\)	Common side
5. \(\triangle ABC\cong\triangle CDA\)	SAS (Steps 2, 3, 4)

For the second proof ( \(\triangle ABD\cong\triangle CBD\)):

Statement	Reason
2. \(AD = CD\)	Given
3. \(BD = BD\)	Common side
4. \(\triangle ABD\cong\triangle CBD\)	SSS (Steps 1, 2, 3)

If we consider the first proof:

Step 1: Identify alternate interior angles

Since \(AB\parallel DC\), \(\angle BAC=\angle DCA\) (alternate interior angles theorem).

Step 2: List equal sides

We know \(AB = DC\) (given) and \(AC = AC\) (common side).

Step 3: Apply SAS criterion

By SAS (\(AB = DC\), \(\angle BAC=\angle DCA\), \(AC = AC\)), \(\triangle ABC\cong\triangle CDA\).

Answer:

\(\triangle ABC\cong\triangle CDA\) (by SAS) and \(\triangle ABD\cong\triangle CBD\) (by SSS)