85.5k views
2 votes
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

User Kasperd
by
7.7k points

1 Answer

5 votes

Final answer:

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'.

User Scott McLeod
by
8.1k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories