Question

Triangle A C D is shown. Point B is near the center of the triangle. Angle A C D is bisected by line segment C B. Lines are drawn from point B to points A and D to form 3 triangles within the larger triangle. Angle C B D is 125 degrees. The length of line segment A C is 52 centimeters and the length of line segment B D is 29 centimeters.
If Line segment C B. bisects ∠ACD, what additional information could be used to prove ΔABC ≅ ΔDBC using SAS? Select three options.

m∠ABC = 125° and AB ≅ DB
ΔACD is isosceles with base AD
ΔABD is isosceles with base AD
CD = 52 cm
AB = 29 cm

Answer 1

Selected:
- m∠ABC = 125° and AB ≅ DB
- ΔACD is isosceles with base AD
- CD = 52 cm

Reasons (brief):
- If m∠ABC = 125° and AB ≅ DB, then ∠ABC = ∠DBC (both 125°), BC is common, and AB = DB → SAS.
- If ΔACD is isosceles with base AD, then AC = CD. Since CB bisects ∠ACD, ∠ACB = ∠BCD, and BC is common → AC = CD with included equal angles gives SAS.
- If CD = 52 cm (with AC = 52 cm given), then AC = CD; with CB bisecting ∠ACD (so ∠ACB = ∠BCD) and BC common → SAS.

The other two options alone do not provide the needed equal included angles for SAS.