Question

question 9
triangle abc, shown in the diagram below, is an isosceles triangle.
if the measure of ∠acd is 140°, what is the measure of ∠cab?
o 140°
o 100°
o 80°
o 40°

Explanation:

Step1: Find ∠ACB

Since ∠ACD + ∠ACB = 180° (linear - pair of angles), and ∠ACD = 140°, then ∠ACB=180° - 140° = 40°.

Step2: Use isosceles - triangle property

In isosceles triangle ABC with AC = AB, ∠ACB = ∠ABC = 40°.

Step3: Calculate ∠CAB

Using the angle - sum property of a triangle (∠CAB+∠ABC + ∠ACB = 180°), we substitute ∠ABC = 40° and ∠ACB = 40°. So, ∠CAB=180°-(40° + 40°)=100°. But we made a mistake above. Since ∠ACD is an exterior angle of triangle ABC. By the exterior - angle property of a triangle, ∠ACD=∠CAB + ∠ABC. In isosceles triangle ABC with AC = AB, ∠ACB = ∠ABC. Let ∠CAB=x and ∠ABC = ∠ACB = y. We know ∠ACD = 140°. And ∠ACD=∠CAB + ∠ABC (exterior - angle property). Also, in △ABC, x + 2y=180°. Since ∠ACD = 140°, then 140°=x + y. From x + 2y=180° and x + y=140°, we subtract the second equation from the first: (x + 2y)-(x + y)=180° - 140°, which gives y = 40°. Substitute y = 40° into x + y=140°, we get x = 80°. So ∠CAB = 80°.

Answer:

C. 80°