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, blank. angles jgm and khm are right angles. thus, blank congruent by the vertical angles theorem. in congruent triangles, corresponding sides are congruent, thus jg ≅ kh. in conclusion jg = kh by blank. then, it follows that △jmg ≅ △kmh by the blank because all right angles are congruent. it is also given that blank by the definition of a midpoint. since vertical angles are congruent by the definition of congruent angles.

Explanation:

Step1: Use mid - point property

Since M is the mid - point of segment GH, we have GM = MH by the definition of a mid - point.

Step2: Identify angle congruence

Given that ∠JGM and ∠KHM are right angles, so ∠JGM=∠KHM = 90°. Also, ∠JMG and ∠KMH are vertical angles, and vertical angles are congruent, so ∠JMG = ∠KMH.

Step3: Apply congruence criterion

By the Angle - Side - Angle (ASA) congruence criterion, in △JMG and △KMH, we have ∠JGM = ∠KHM, GM = MH, and ∠JMG = ∠KMH. So, △JMG≅△KMH.

Step4: Use property of congruent triangles

Since △JMG≅△KMH, by the property that corresponding sides of congruent triangles are congruent, we have JG = KH.

Answer:

The proof is completed by first using the mid - point property to get GM = MH, then identifying angle congruences (right angles and vertical angles), applying the ASA congruence criterion to show △JMG≅△KMH, and finally using the property of congruent triangles to conclude JG = KH.