Refer to the triangle for numbers 8 – 19.

(First triangle is here)

The angle bisector theorem state that the ratio of AB/AD is equal to the ratio of BC/CD. Is the ratio of (AB+BC)/(AD+DC) the same?

First, extend line BC. Then draw a line from point A, parallel to segment BD and intersecting the extended BC. The point of intersection will be E. The image will appear like the one below.

8. Classify angle AEC and angle DBCas corresponding, alternate interior, alternate exterior, or same side interior angles.

(Second triangle here)

9. When lines are parallel, are angles AEC and DBC congruent or supplementary?

10. In ∆AEC and ∆DBC angle C is shared. What property confirms that angle C will be congruent in both triangles?

11. Which similarity postulate or theorem confirms that ∆AEC~∆DBC

12. Complete the proportion BC/?=?/AC

13. Use the segment addition postulate to rewrite the length of EC.

14. Use the substitution property and plug the answer to number 13 into the proportion in number 12

15. Classify angle BAE and angle DBA as corresponding, alternate interior, alternate exterior, or same side interior angles.

16. When lines are parallel, are angles BAE and DBA congruent or supplementary?
17. Since it was given that angles DBA and DBC were congruent, which property can be used to say angles BAE and AEC are also congruent?

18. Classify ∆ABE as isosceles, scalene, or equilateral.

19. Use the substitution property and the proportion in number 14 to confirm the ratios at the start are the same.