.In <ABC, <A=60,<B=70,<C=50. Points D , E ,F are the mid-pointsmof the sides BC ,AC and AB respectively. Find the measures of the angles of the triangle formed by joining the mid-points of the sides of <ABC.
since the triangles are similar, the angles are all the same as in ABC.
To find the measures of the angles of the triangle formed by joining the midpoints of the sides of triangle ABC, we can use the fact that the line joining the midpoints of two sides of a triangle is parallel to the third side and divides it into two equal segments.
Given that D, E, and F are the midpoints of sides BC, AC, and AB respectively, we can use this information to find the measures of the angles of the triangle formed by joining these midpoints.
Let's label the triangle formed by joining the midpoints as XYZ, with X corresponding to segment DE, Y corresponding to segment EF, and Z corresponding to segment FD.
Since DE is parallel to AB and BC, the angles at X and Y will be equal to the angles at B and C respectively. Similarly, the angle at Z will be equal to the angle at A.
To find the measure of the angles, we need to determine the corresponding measures of angle B and angle C in triangle ABC.
We are given that angle A = 60 degrees, angle B = 70 degrees, and angle C = 50 degrees.
Since angles B and C are angles of triangle ABC, they will be equal to the corresponding angles in triangle XYZ.
Therefore, angle X = angle B = 70 degrees and angle Y = angle C = 50 degrees.
To find angle Z, we subtract the sum of angles X and Y from 180 degrees (since the angles of a triangle add up to 180 degrees).
angle Z = 180 - (angle X + angle Y)
= 180 - (70 + 50)
= 180 - 120
= 60 degrees
Therefore, the measures of the angles of the triangle formed by joining the midpoints of the sides of triangle ABC are:
angle X = 70 degrees
angle Y = 50 degrees
angle Z = 60 degrees