Question

1. which pair of triangles are congruent based on the given information?

Explanation:

Step1: Analyze the first pair of triangles

In the first pair of triangles (with \(BD\perp AC\)), we only know that \(\angle ADB=\angle CDB = 90^{\circ}\) and \(\angle A=\angle C\), but we don't have information about the side - lengths. So, we cannot prove congruence using any of the congruence criteria (ASA, AAS, SSS, SAS, HL).

Step2: Analyze the second pair of triangles

For \(\triangle EFG\) and \(\triangle JHG\), we know \(\angle E=\angle J\) (given), \(\angle F=\angle H\) (given). But we don't know if any corresponding sides are equal. So, we cannot prove congruence using ASA or AAS (since we lack side - length information).

Step3: Analyze the third pair of triangles

For \(\triangle BDM\) and \(\triangle HDM\):

• We have \(\angle BDM=\angle HDM = 55^{\circ}\) (given).
• \(DM = DM\) (common side).
• \(\angle BMD=\angle HMD=60^{\circ}\) (given).

By the ASA (Angle - Side - Angle) congruence criterion (\(\angle BDM=\angle HDM\), \(DM = DM\), \(\angle BMD=\angle HMD\)), \(\triangle BDM\cong\triangle HDM\).

Answer:

The pair of triangles \(\triangle BDM\) and \(\triangle HDM\) are congruent.