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, because all right angles are congruent. since vertical angles are congruent by the vertical angles th. 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: Congruent Segments

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

Step2: Congruent Right Angles

Given \(\angle JGM\) and \(\angle KHM\) are right angles. So, \(\angle JGM\cong\angle KHM\) (because all right angles are congruent).

Step3: Congruent Vertical Angles

Since \(\angle JMG\) and \(\angle KMH\) are vertical angles, \(\angle JMG\cong\angle KMH\) (by the vertical angles theorem).

Step4: Triangle Congruence Criterion

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

Answer:

The second blank: \(\angle JGM\cong\angle KHM\); The third blank: \(AAS\)