Question

In triangle ABC, angle bisector BM and altitude BD are drawn. In triangle BMC, altitude MK is drawn. The angle between BM and BD is 20^circ, and the angle between BM and MK is 50^circ. What is the ratio of the external angle measures, in lowest terms?

a) 1:3
b) 1:2
c) 2:3
d) 3:4

Answer 1

Calculating the internal and external angles of triangle BMC, we find the external angle at vertex C is 40°. Comparing this to the given 20° angle at B, the ratio of the external angle measures is 1:2.

Step-by-step explanation:

Let's consider triangle BMC with an angle of 50° between BM and MK. BM is the angle bisector, and thus it bisects angle BMC into two equal angles. The angle between BD and BM is given as 20°, which is one of these angles. This implies that angle CBM also measures 20° because BM is the angle bisector. Therefore, the remaining angle, angle BCM, is 140° since the sum of angles in a triangle is 180° (50° + 20° + 20°).

Knowing that angle BCM is 140°, we can find the measure of the external angle at vertex C by subtracting the internal angle BCM from 180° which gives us 40° as the external angle measurement for vertex C. Now, we note that this external angle is an extension of the internal angle at B in triangle ABC, and therefore, the external angle ratio for B and C respectively becomes 20° : 40°, which simplifies to 1:2, which is option 'b'.