Question

determine if the triangles are congruent. explain your reasoning.
1.
2.
3.
4.

Explanation:

Problem 1:

Step1: Identify Angles and Sides

In $\triangle ABC$ and $\triangle DEF$:

• $\angle C = 45^\circ$, $\angle B = 25^\circ$, side $BC = 8$ mm.
• $\angle F = 45^\circ$, $\angle E = 25^\circ$, side $ED = 8$ mm.

Step2: Check Congruence Criterion

Using the ASA (Angle - Side - Angle) or AAS (Angle - Angle - Side) criterion:

• Two angles and a non - included side (or included side) are equal.
• $\angle C=\angle F = 45^\circ$, $\angle B=\angle E = 25^\circ$, and the side between the angles (or a corresponding side) $BC = ED=8$ mm.
• By AAS congruence criterion, $\triangle ABC\cong\triangle DEF$.

Step1: Identify Angles and Sides

In $\triangle QSR$ and $\triangle UTV$:

• $\angle S=\angle V = 90^\circ$ (right angles).
• Side $QS = TV = 4$ in.
• $\angle Q=\angle T$ (corresponding angles).

Step2: Check Congruence Criterion

Using the ASA (Angle - Side - Angle) congruence criterion:

• $\angle S=\angle V$, $QS = TV$, $\angle Q=\angle T$.
• So, $\triangle QSR\cong\triangle UTV$ by ASA.

Step1: Identify Angles and Sides

In $\triangle HLJ$ and $\triangle HKJ$:

• $HJ$ is common to both triangles.
• $HL = HK=2.2$ ft.
• $\angle L=\angle K = 63^\circ$.

Step2: Check Congruence Criterion

Using the SAS (Side - Angle - Side) congruence criterion:

• $HL = HK$, $\angle L=\angle K$, $HJ = HJ$ (common side).
• So, $\triangle HLJ\cong\triangle HKJ$ by SAS.

Answer:

The triangles are congruent by AAS (Angle - Angle - Side) congruence criterion.

Problem 2: