Question

use the diagram.
given: \\(\overline{ab} \cong \overline{ad}; \overline{bc} \cong \overline{dc}\\)
prove: \\(\angle abc \cong \angle adc\\)

statements	reasons
2. \\(\overline{ac} \cong \overline{ac}\\)	2. reflexive property of congruence
3. \\(\triangle abc \cong \triangle adc\\)	3.
4. \\(\angle abc \cong \angle adc\\)	4.

Explanation:

Step1: Identify Triangle Congruence

We have \( \overline{AB} \cong \overline{AD} \), \( \overline{BC} \cong \overline{DC} \), and \( \overline{AC} \cong \overline{AC} \) (reflexive property). By SSS (Side - Side - Side) congruence criterion, \( \triangle ABC \cong \triangle ADC \).

Step2: Corresponding Angles Congruence

Since \( \triangle ABC \cong \triangle ADC \), their corresponding angles \( \angle ABC \) and \( \angle ADC \) are congruent (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Filling the Table:

• For row 3 (Reason for \( \triangle ABC \cong \triangle ADC \)): The reason is "SSS Congruence Criterion" (because we have three pairs of congruent sides: \( \overline{AB} \cong \overline{AD} \), \( \overline{BC} \cong \overline{DC} \), \( \overline{AC} \cong \overline{AC} \)).
• For row 4 (Reason for \( \angle ABC \cong \angle ADC \)): The reason is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Answer:

• Row 3 Reason: SSS Congruence Criterion
• Row 4 Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)