Question

select the correct answer from each drop - down menu.
given: m is the midpoint of segment gh.
angles jgm and khm are right angles.
prove: ( jg = kh )
complete the proof.
it is given that m is the midpoint of segment gh. therefore by the definition of a midpoint. it is also given that angles jgm and khm are right angles. thus, cause all right angles are congruent. since vertical angles are congruent by the vertical angles the. then, it follows that ( \triangle jmgcong\triangle kmh ) by the criteria. in congruent, thus ( overline{jg}congoverline{kh} ). in conclusion ( jg = kh ) by definition of congruent angles.

Explanation:

Step1: Mid - point property

Since \(M\) is the mid - point of \(GH\), by the definition of a mid - point, \(GM = HM\).

Step2: Right - angle congruence

Angles \(JGM\) and \(KHM\) are right angles. So, \(\angle JGM=\angle KHM = 90^{\circ}\) (all right angles are congruent).

Step3: Vertical - angle congruence

\(\angle JMG\) and \(\angle KMH\) are vertical angles. By the vertical - angles theorem, \(\angle JMG=\angle KMH\).

Step4: Triangle - congruence criterion

In \(\triangle JMG\) and \(\triangle KMH\), we have \(\angle JGM=\angle KHM\), \(GM = HM\), and \(\angle JMG=\angle KMH\). So, \(\triangle JMG\cong\triangle KMH\) by the \(ASA\) (Angle - Side - Angle) criterion.

Step5: Congruent - segment property

Since \(\triangle JMG\cong\triangle KMH\), their corresponding sides are congruent. So, \(JG = KH\) (by the definition of congruent triangles, corresponding parts of congruent triangles are congruent).

Answer:

The first drop - down: segment \(GM\) is congruent to segment \(HM\); the second drop - down: \(ASA\)