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In the diagram below, AM = BM = CM and angle BMC + angle A  = 201 degrees.  Find angle B in degrees. Aug 6, 2021

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angle BMC + angle A = 201°

angle BMC = 201° - angle A  (eq. 1)

angle BMC + angle AMC = 180° (straight line at point M)

angle AMC = 180° - angle BMC

angle AMC = 180° - (201° - angle A) (by substituting from eq. 1 above for angle BMC)

angle AMC = angle A - 21° (eq. 2)

If AM = CM, then angle A = angle ACM (eq. 3)

The three angles for triangle ACM add to 180°:

180° = angle A + angle ACM + angle AMC

180° = angle A + angle A + angle A - 21° (by substituting from eq. 2 and eq. 3 from above for angle AMC and angle ACM respectively)

180° = 3 angle A - 21°

201° = 3 angle A

angle A = 67°

If BM = CM, then angle B = angle BCM (eq. 4)

The three angles for triangle ABC add to 180°:

180° = angle A + angle ACM + angle BCM + angle B (where we have used the fact that angle C = angle ACM + angle BCM)

180° = angle A + angle A + angle B + angle B (by substituting from eq. 3 and eq. 4 for angle ACM and angle BCM respectively)

180° = 2 angle A + 2 angle B

90° = angle A + angle B

angle B = 90° - angle A

angle B = 90° - 67°

angle B = 23°

Aug 6, 2021