Explanation:
prove the triangle proportionality theorem.
answer:-
If a line parallel to one side of a triangle intersects the other two sides of the triangle, then that line divides these two sides proportionally.
From the statement: If ��∣∣��FG∣∣BC then,
Show that: ����=����FAFB=AGGC
Consider △���△ABC and △���△GFA
Reflexive property states that the value is equal to itself.
∠���≅∠���∠BAC≅∠GAF [Angle] {Reflexive property of equality}
Corresponding angles theorem states that if the two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent(i., e equal).
∠���≅∠���∠ABC≅∠GFA [Angle]
∠���≅∠���∠ACB≅∠AGF [Angle]
AA Similarity states that the two triangles have their corresponding angles equal if and only if their corresponding sides are proportional.
then, by AA similarity theorem:
△���∼△���△ABC∼△GFA
By segment addition postulates:
AB = FA +FB and AC = AG + GC
Corresponding sides in similar triangles are proportional
����=����FAAB=AGAC .....[1]
Substitute AB = FA +FB and AC = AG + GC in [1]
we have;
��+����=��+����FAFA+FB=AGAG+GC
Separate the fraction:
����+����=����+����FAFA+FAFB=AGAG+AGGC
Simplify:
1+����=1+����1+FAFB=1+AGGC
Subtract 1 from both sides we get;
����=����FAFB=AGGC hence proved