Home › Geometry - angles and clocks
Find an angle using a bisector and a straight line
A bisector halves an angle; adjacent angles on a line total 180 degrees
If a ray bisects an angle, both parts are equal. If two adjacent angles make a straight line, their measures add to 180∘.
Problem
Point M lies on the straight line AB. Ray MD bisects angle CMB, and angle DMC is 55∘. Find angle CMA.
Answer
∠CMA=70∘.
The requested angle is adjacent to the whole angle $CMB$, not to just one half of it.
Step-by-step solution
1. Rebuild the bisected angle
Because MD bisects ∠CMB, the two parts are equal: ∠DMC=∠DMB=55∘. Therefore ∠CMB=55∘+55∘=110∘.
2. Use the straight line
Rays MA and MB point in opposite directions. Thus ∠CMA+∠CMB=180∘, giving ∠CMA=180∘−110∘=70∘.
Geometry Solver