In parallelogram ABCD,M is the mid point of diagonal BD;if BM bisects angle ABC,then what is the measurement of angle AMB?
90°
To find the measurement of angle AMB, we can use the properties of a parallelogram. Here's how you can find the answer step by step:
1. Draw parallelogram ABCD and label the given points: M is the midpoint of diagonal BD, and BM bisects angle ABC.
2. Since M is the midpoint of diagonal BD, we can conclude that BM is half the length of BD. This means that triangle BMD is congruent to triangle AMD.
3. Since triangle BMD is congruent to triangle AMD, their corresponding angles are equal. Therefore, angle BMD is equal to angle AMD.
4. Since BM bisects angle ABC, we can conclude that angle MBM is equal to angle ABC. This follows from the property that if a line divides an angle into two congruent angles, it is called an angle bisector.
5. Now, consider triangle AMB. We know that angle BMD is equal to angle AMD (from step 3) and angle MBM is equal to angle ABC (from step 4).
6. By the angle sum property of a triangle, the sum of all angles in a triangle is 180 degrees. Therefore, the measurement of angle AMB can be found by subtracting angles BMD and MBM from 180 degrees.
Angle AMB = 180 degrees - angle BMD - angle MBM
Substituting the equal angles we found in steps 3 and 4:
Angle AMB = 180 degrees - angle AMD - angle ABC
Since angle BMD is equal to angle AMD and angle MBM is equal to angle ABC:
Angle AMB = 180 degrees - angle BMD - angle MBM
Angle AMB = 180 degrees - angle BMD - angle BMD (substituting equal angles)
Angle AMB = 180 degrees - 2 * angle BMD
Now, you can find the measurement of angle AMB by knowing the value of angle BMD.