I hope this message finds you well. I am currently working on a geometric problem that involves finding the value of a specific angle($∠CAD$). While I have already solved it using the sine and cosine laws, I am eager to explore alternative methods and derive the solution without relying on these laws. I have already solved the value is $30°$.
I kindly request your assistance in exploring alternative approaches to solve this problem. I believe that employing different mathematical concepts or principles may make it possible to arrive at the desired angle without invoking the sine and cosine laws. Your expertise and insights in this matter would be greatly appreciated.
To provide you with some context, here is a brief description of the problem:
In the above triangle, the angle $∠BAD= 15°$ & $∠ADC=45°$. Find the value of $∠CAD$, when $CD=BD$
Thank you in advance for your time and assistance. I look forward to your valuable insights and guidance.
Best Answer
Let $E \in AB$ such that $CE\perp AB$. Then $\triangle CED$ is equilateral (because $ED \cong CD$ and $\measuredangle ECD = 60^\circ$); as a consequence, $\measuredangle EDA = 15^\circ$. Therefore $\triangle AED$ is isosceles, and $AE \cong ED \cong EC$. We have thus $\measuredangle EAC = 45^\circ$. Hence $$\boxed{\measuredangle CAD = 30^\circ}.$$