Question

determine if the two triangles are necessarily congruent. if so, fill in a flowchart proof to prove that they are.

the triangles are necessarily congruent.

for each of the first three boxes, choose a statement format from the dropdown menu. you will then be able to change the letters to match the diagram for this problem.

Explanation:

Step1: Identify Given Information

In $\triangle ZAB$ and $\triangle NLM$:

• Side $ZA \cong$ Side $LM$ (marked with one tick).
• Side $ZB \cong$ Side $NM$ (marked with two ticks).
• Angle $\angle Z \cong \angle N$ (marked with arc).

Step2: Apply SAS Congruence Criterion

The SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Here, we have two sides ( $ZA \cong LM$, $ZB \cong NM$) and the included angle ( $\angle Z \cong \angle N$) congruent between $\triangle ZAB$ and $\triangle NLM$ (or $\triangle BAZ$ and $\triangle LMN$ as per the diagram's labeling for the congruence statement). So by SAS, the triangles are congruent.

Step3: Fill the Flowchart Statements

• First box (Statement): $ZA \cong LM$; Reason: GIVEN
• Second box (Statement): $ZB \cong NM$; Reason: GIVEN
• Third box (Statement): $\angle Z \cong \angle N$; Reason: GIVEN
• Final box (Statement): $\triangle LMN \cong \triangle BAZ$; Reason: SAS Congruence Criterion

Answer:

The triangles are necessarily congruent by the SAS (Side - Angle - Side) congruence criterion. The flowchart statements are filled as:

1. Statement: $ZA \cong LM$, Reason: GIVEN
2. Statement: $ZB \cong NM$, Reason: GIVEN
3. Statement: $\angle Z \cong \angle N$, Reason: GIVEN
4. Statement: $\triangle LMN \cong \triangle BAZ$, Reason: SAS Congruence Criterion