Chapter 4: Q13. (page 137)

Write proofs in two–column form.
Given: $M$ is the midpoint of $\bar{JK}$ ; $∠ 1 ≅ ∠ 2$
Prove: $\bar{JG} ≅ \bar{MK}$

Expert verified

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

Step by step solution

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 $\bar{J G}$ is opposite to $∠ 2$ and $\bar{J M}$ is opposite to $∠ 1$ .

As $∠ 1 ≅ ∠ 2$ , $\bar{J G} ≅ \bar{J M}$ .

Since $M$ is the midpoint of $\bar{J K}$ , by definition of midpoint

$\bar{J M} ≅ \bar{M K}$

If $∠ P ≅ ∠ Q$ and $∠ Q ≅ ∠ S$ then $∠ P ≅ ∠ S$ .

Since, $\bar{J M} ≅ \bar{M K}$ and $\bar{J G} ≅ \bar{J M}$ implies that $\bar{J G} ≅ \bar{M K}$ .

Write a two-column proof based on above explanation.

Statement	Reason
$∠ 1 ≅ ∠ 2$	Given
$\bar{J G} ≅ \bar{J M}$	Converse of isosceles theorem
$M$ is the midpoint of $\bar{J K}$	Given
$\bar{J M} ≅ \bar{M K}$	Midpoint definition
$\bar{J G} ≅ \bar{M K}$	Transitive property