Question

Write proofs in two–column form.

Given:$M$ is the midpoint of $\overline{\mathrm{JK}}$;$\angle \mathbf{1}\cong \angle \mathbf{2}$

Prove:$\overline{\mathrm{JG}}\cong \overline{\mathrm{MK}}$

Short answer

Statement	Reason
$\angle \mathbf{1}\cong \angle \mathbf{2}$	Given
$\overline{\mathrm{JG}}\cong \overline{\mathrm{JM}}$	Converse of isosceles theorem
$M$is the midpoint of$\overline{\mathrm{JK}}$	Given
$\overline{\mathrm{JM}}\cong \overline{\mathrm{MK}}$	Midpoint definition
$\overline{\mathrm{JG}}\cong \overline{\mathrm{MK}}$	Transitive property

Step-by-step solution

Step 1. Converse of isosceles theorem.

If two angles of a triangle are congruent, then the sides opposite to those angles are congruent.

From the given figure it can be observed that$\overline{JG}$ is opposite to$\angle 2$ and$\overline{JM}$ is opposite to$\angle 1$.

As $\angle 1\cong \angle 2$, $\overline{JG}\cong \overline{JM}$.

Step 2. Description of step.

Since $M$is the midpoint of $\overline{JK}$, by definition of midpoint

$\overline{JM}\cong \overline{MK}$

Step 3. Transitive property of congruence.

If$\angle P\cong \angle Q$ and$\angle Q\cong \angle S$ then $\angle P\cong \angle S$.

Since, $\overline{JM}\cong \overline{MK}$and $\overline{JG}\cong \overline{JM}$implies that $\overline{JG}\cong \overline{MK}$.

Step 4. Two column proof.

Write a two-column proof based on above explanation.

Statement	Reason
$\angle 1\cong \angle 2$	Given
$\overline{JG}\cong \overline{JM}$	Converse of isosceles theorem
$M$is the midpoint of$\overline{JK}$	Given
$\overline{JM}\cong \overline{MK}$	Midpoint definition
$\overline{JG}\cong \overline{MK}$	Transitive property