MATH SOLVE

5 months ago

Q:
# Given: Triangles ABC and DBC are isosceles, m∠BDC = 30°, and m∠ABD = 155°.Find m∠ABC, m∠BAC, and m∠DBC.

Accepted Solution

A:

mâ ABC = 35Â°
mâ BAC = 72.5Â°
mâ DBC = 120Â°
Since the diagram wasn't provided, I am making the following assumption:
â DBC and â ABC are the vertex angles of the two isosceles triangles.
Since, mâ BDC = 30Â° and triangle DBC is isosceles, then mâ BCD = 30Â° as well, and mâ DBC = 180Â° - 30Â° - 30Â° = 120Â°
Since mâ ABD = 155Â°, and mâ ABD = mâ DBC + mâ ABC, that means that mâ ABC = â ABD - â DBC = 155Â° - 120Â° = 35Â°
Finally, since mâ ABC = 35Â° and triangle ABC is isosceles, then mâ BAC = (180Â° - 35Â°)/2 = (145Â°)/2 = 72.5Â°
Pay attention to the assumption in the above calculations. There is a total of 9 different possibilities for this question depending upon what angles are the vertex angles of the triangles. (Actually, only 7 of the 9 potential vertex angles work. 2 of the 9 have the base angles exceed 90 degrees which is impossible for a triangle).
Vertex angles
â ABC & â DBC: mâ ABC = 35Â°, mâ BAC = 72.5Â°, mâ DBC = 120Â°
â ABC & â BDC: mâ ABC = 80Â°, mâ BAC = 50Â°, mâ DBC = 75Â°
â ABC & â DCB: mâ ABC = 125Â°, mâ BAC = 27.5Â°, mâ DBC = 30Â°
â ACB & â DBC: mâ ABC = 35Â°, mâ BAC = 35Â°, mâ DBC = 120Â°
â ACB & â BDC: mâ ABC = 80Â°, mâ BAC = 80Â°, mâ DBC = 75Â°
â ACB & â DCB: Impossible
â BAC & â DBC: mâ ABC = 35Â°, mâ BAC = 110Â°, mâ DBC = 120Â°
â BAC & â BDC: mâ ABC = 80Â°, mâ BAC = 20Â°, mâ DBC = 75Â°
â BAC & â DCB: Impossible