Question

based on the information provided, determine whether a congruence exists between the two triangles. if a congruence exists between the triangles, state the congruences and the criteria used to determine them. given: m is the mid - point of hp, m∠h = m∠p

Explanation:

Step1: Recall mid - point property

Since M is the mid - point of $\overline{HP}$, we have $HM = MP$.

Step2: Identify vertical angles

$\angle HMG$ and $\angle PMR$ are vertical angles, so $\angle HMG=\angle PMR$.

Step3: Use given angle equality

We are given that $\angle H=\angle P$.

Step4: Apply ASA congruence criterion

In $\triangle HMG$ and $\triangle PMR$, we have $\angle H=\angle P$, $HM = MP$, and $\angle HMG=\angle PMR$. By the Angle - Side - Angle (ASA) congruence criterion, $\triangle HMG\cong\triangle PMR$.

Answer:

A congruence exists. $\triangle HMG\cong\triangle PMR$ by the ASA (Angle - Side - Angle) criterion.