Notice that FL and HN are both made from the bases of two isosceles triangles.
See solution.
Practice makes perfect
Examining the diagram, we see that FL and HN are both made from the bases of two isosceles triangles. By the Segment Addition Postulate, we can write the following equations.
FL&=FJ+JL
HN&=HK+KN
If we can prove the following, we will have enough information to show that FL≅ HN.
Show:
△ FGJ≅ △ HGK and △ JML≅ △ KMN
Examining the diagram, we see that the vertex angles of △ FGJ and △ HGK are vertical angles. This is also true for the vertex angles of △ JML and △ KMN. According to the Vertical Angles Congruence Theorem, vertical angles are congruent. Let's show this in the diagram.
Since two sides and the included angle of △ FGJ are congruent to two sides and the included angle of △ HGK, we know these triangles are congruent by the SAS Congruence Theorem. Using the same theorem, we can also prove that △ JML and △ KMN are congruent. Let's mark the last corresponding sides in each of the triangles.
Since FJ≅ HK and JL≅ KN it follows by the Segment Addition Postulate and the definition of congruence that the given statement is true.
FL≅ HN
Proof
Two-Column Proof
Let's show this as a two column proof.
Statement
Reason
1.
&FG≅ GJ≅ HG≅ GK &JM≅ LM≅ KM≅ NM
1.
Given
2.
& ∠ FGJ≅ ∠ HGK &∠ JML ≅ ∠ KMN
2.
Vertical Angles Congruence Theorem
3.
△ FGJ ≅ △ HGK △ JML ≅ △ KMN
3.
SAS Congruence Theorem
4.
FJ ≅ HK JL ≅ KN
4.
Corresponding parts of congruent triangles are congruent
