Question

Segments AB , CD , and EF intersect at point O, points A, E, C and points B, F, D are collinear so that AO ≅ OB , CO ≅ OD . Prove that AE ≅ BF .

Answer 1

If points A, E and C are colinear, then they lie on the same line. The same statement you can say about points B, F and D.

1. Consider triangles AOC and BOD. In these triangles:

• AO≅OB (given);
• CO≅OD (given);
• ∠AOC≅∠BOD (as vertical angles).

Thus, ΔAOC≅ΔBOD by SAS Postulate (If any two corresponding sides and their included angle are the same in both triangles, then the triangles are congruent). Corresponding parts of congruent triangles are congruent, then

• AC≅BD;
• ∠ACO≅∠BDO;
• ∠CAO≅∠DBO.

Since angles ACO and BDO are alternate interior angles between lines AE and BF with transversal CD and these angles are congruent, then lines AE and BF are parallel.

This gives you that

• ∠CEO≅∠OFD;
• ∠ECO≅∠ODF.

2. Consider triangles ECO and FDO. In these triangles

• ∠CEO≅∠OFD (previous proof);
• CO≅OD (given);
• ∠ECO≅∠ODF (previous proof).

Therefore, ΔECO≅ΔFDO by AAS Postulate (if two angles and the non-included side one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent). Then CE≅FD.

3. Note that

• AE=AC+CE;
• BF=BD+DF.

Since AC≅BD and CE≅DF, then AE=AC+CE=BD+DF=BF.