Geometry Theorems

Thales' Theorem and the Angle Bisector Theorem

Thales' Theorem (Basic Proportionality Theorem)

Key Concept: If a line is drawn parallel to one side of a triangle, intersecting the other two sides, it divides them proportionally.

Practice Problem

In ∆ABC, DE || BC. If AD = 6 cm, DB = 4 cm, and AE = 9 cm, find EC.

Answer: EC = 6 cm

Angle Bisector Theorem

Key Concept: The angle bisector of a triangle divides the opposite side into segments proportional to the adjacent sides.

Practice Problem

In ∆DEF, DG bisects ∠D. If DE = 10 cm, DF = 14 cm, and EF = 12 cm, find EG and GF.

Answer: EG = 5 cm, GF = 7 cm