In the given figure AD = AB = AC BD is parallel to CA a
![In the given figure AD = AB = AC BD is parallel to CA a](https://static.tllms.com/ckeditor_assets/pictures/259993/content_802061.jpg)
In the given figure, AD = AB = AC, BD is parallel to CA and angle ACB = 65°. The measure of ∠DAC is
A. 50∘
B. 80∘
C. 130∘
D. 150∘
Please scroll down to see the correct answer and solution guide.
Right Answer is: C
SOLUTION
We can see that △ABC is an isoceles triangle with side AB = side AC.
⟹∠ACB=∠ABC (In a triangle, if two sides are equal, then the angles opposite to these sides are also equal).
As ∠ACB=65∘, we must have ∠ABC=65∘
We know that, the sum of all the angles of a triangle is 180∘.
∠ACB+∠CAB+∠ABC=180∘
⟹65∘+65∘+∠CAB=180∘
⟹∠CAB = 180∘−130∘
i.e., ∠CAB=50∘
As BD is parallel to CA, we must have, ∠CAB=∠DBA, since they are alternate angles.
∠CAB=∠DBA=50∘
We see that △ADB is an isosceles triangle with Side AD = Side AB.
⟹∠ADB=∠DBA=50∘
Since sum of all the angles of a triangle is 180°,
∠ADB+∠DAB+∠DBA=180°
⟹50∘+∠DAB+50∘=180∘
⟹∠DAB=180∘−100∘=80∘
⟹∠DAB=80∘
The angle DAC is sum of angle DAB and CAB.
∠DAC=∠CAB+∠DAB
⟹∠DAC=50∘+80∘
i.e., ∠DAC=130∘