We are given 4 pairs of congruent segments, and we need to prove that ∠ G ≅ ∠ 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.
GL &= GH + HLHowever, since GH≅JH and HL≅HM, we get GH = JH and HL = HM. Let's substitute them into the equation above.
GL &= JH + HM_(JM)
From the equation above, we conclude that GL ≅ JM. Next, we will separate the triangles △ PGL and △ KLM.
One more time, we apply the Segment Addition Postulate and write the following relation.
PL = PM + ML
Since PM ≅ KL we get PM= KL. Let's substitute it into the equation above.
PL = PM + ML ⇒ PL &= KL+ ML
PL &= KM
From the latter equation we conclude that PL≅ KM. Consequently, by the Side-Side-Side (SSS) Congruence Postulate we have △ PGL ≅ △ KJM and so, by definition, ∠ G ≅ ∠ J.
Completed Proof
Given: & HL ≅ HM, PM ≅ KL
& PG ≅ KJ, GH ≅ JH
Prove: & ∠ G ≅ ∠ J
Proof: To prove that ∠ G ≅ ∠ J, it is enough to show that △ PGL ≅ △ 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 GH ≅ JH and HL ≅ HM, we have GH=JH and HL=HM. By substituting them into the initial equation we get GL=JH+HM=JM and then GL≅ JM.
Similarly, by the Segment Addition Postulate we have PL=PM+ML, but since PM≅ LK, we get PM=LK. Substituting it into the previous equation we get PL=LK+ML=KM, which implies PL ≅ KM. Remember that we are also given that PG ≅ KJ.
