Exercise 26 Page 271

We are given $4$ pairs of congruent segments, and we need to prove that $\angle G\cong \angle J.$ Let's highlight all this information in the given diagram.

Next, by the diagram and using the Segment Addition Postulate, we can write the following relations. However, since $\overline{GH}\cong \overline{JH}$ and $\overline{HL}\cong \overline{HM},$ we get $GH=JH$ and $HL=HM.$ Let's substitute them into the equation above. From the equation above, we conclude that $\overline{GL}\cong \overline{JM}.$ Next, we will separate the triangles $△PGL$ and $△KLM.$ One more time, we apply the Segment Addition Postulate and write the following relation. Since $\overline{PM}\cong \overline{KL}$ we get $PM=KL.$ Let's substitute it into the equation above. From the latter equation we conclude that $\overline{PL}\cong \overline{KM}.$ Consequently, by the Side-Side-Side (SSS) Congruence Postulate we have $△PGL\cong △KJM$ and so, by definition, $\angle G\cong \angle J.$

Completed Proof

Proof: To prove that $\angle G\cong \angle J,$ it is enough to show that $△PGL\cong △KJM,$ because congruent parts of congruent polygons are congruent. We will prove this congruence in three steps:

• By the Segment Addition Postulate we have $GL=GH+HL,$ but since $\overline{GH}\cong \overline{JH}$ and $\overline{HL}\cong \overline{HM},$ we have $GH=JH$ and $HL=HM.$ By substituting them into the initial equation we get $GL=JH+HM=JM$ and then $\boxed{\overline{GL}\cong \overline{JM}.}$

• Similarly, by the Segment Addition Postulate we have $PL=PM+ML,$ but since $\overline{PM}\cong \overline{LK},$ we get $PM=LK.$ Substituting it into the previous equation we get $PL=LK+ML=KM,$ which implies $\boxed{\overline{PL}\cong \overline{KM}.}$ Remember that we are also given that $\boxed{\overline{PG}\cong \overline{KJ}.}$

• By the Side-Side-Side (SSS) Congruence Postulate we conclude that $△PGL\cong △KJM.$ Consequently, by definition we get $\angle G\cong \angle J.$