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, segment gm is congruent to segment hm by the definition of a midpoint. it is also given that angles jgm and khm are right angles. thus drop - down menu because all right angles are congruent. since vertical angles are congruent by the vertical angles theorem drop - down menu. then, it follows that (\triangle jmgcong\triangle kmh) by the drop - down menu criteria. in congruent triangles, correspo drop - down menu (kh). in conclusion (jg = kh) by definition of congruent angles.
the drop - down options for one of the blanks are: angle jmg is congruent to angle kmh, angle mjg is congruent to angle mkh, angle jgm is congruent to angle khm, angle jmg is congruent to angle khm

Explanation:

Step1: Identify midpoint congruence

Since $M$ is the midpoint of $\overline{GH}$, $\overline{GM} \cong \overline{HM}$ (definition of midpoint).

Step2: Identify right angle congruence

$\angle JGM$ and $\angle KHM$ are right angles, so $\angle JGM \cong \angle KHM$ (all right angles are congruent).

Step3: Identify vertical angle congruence

$\angle JMG$ and $\angle KMH$ are vertical angles, so $\angle JMG \cong \angle KMH$ (vertical angles theorem).

Step4: Prove triangle congruence

$\triangle JMG \cong \triangle KMH$ by ASA (Angle-Side-Angle) criteria.

Step5: Corresponding parts congruence

Corresponding sides of congruent triangles are congruent, so $\overline{JG} \cong \overline{KH}$, meaning $JG=KH$.

Answer:

1. Blank 1: $\overline{GM} \cong \overline{HM}$
2. Blank 2: $\angle JGM \cong \angle KHM$
3. Blank 3: $\angle JMG \cong \angle KMH$
4. Blank 4: ASA
5. Final conclusion: $JG = KH$