Animated Solution for Mathematics - Trigonometry: With the usual notation, in $\Delta ABC$, if $\angle A+\angle B=120^{\circ }$, $a=\sqrt{3}+1$ and $b=\sqrt{3}-1$, then the ratio $\angle A:\angle B$, is :

Welcome, future engineers! Today, we are embarking on a journey through the elegant landscape of triangle geometry. Imagine you are standing in the middle of a vast, open field, and you are tasked with solving the properties of a triangle $\Delta ABC$.

You are given two pieces of information: the sum of two angles, $\angle A+\angle B=120^{\circ }$, and the lengths of two sides, $a=\sqrt{3}+1$ and $b=\sqrt{3}-1$.

Our first step is to uncover the hidden angle, $\angle C$. We know that in any triangle, the sum of the interior angles is always $180^{\circ }$.

Since we are given $\angle A+\angle B=120^{\circ }$, it follows immediately that:

This is our first victory! We have successfully reduced the complexity of the problem by identifying a fixed point in our geometric puzzle.

Now that we have $\angle C=60^{\circ }$, we have a triangle where we know two sides and the included angle. This is the perfect scenario for a powerful, often underutilized tool in our arsenal: Napier's Analogy, also known as the Tangent Rule.

Why do we choose this? Because it provides a direct, elegant bridge between the difference of the angles and the difference of the sides. The formula is:

This formula is a masterpiece of symmetry. It allows us to bypass the messy calculations of the Sine Rule and jump straight to the relationship between $\angle A$ and $\angle B$.

Let's prepare our terms with precision. We have $a=\sqrt{3}+1$ and $b=\sqrt{3}-1$.

Calculating the difference, $a-b$, gives us $(\sqrt{3}+1)-(\sqrt{3}-1)=2$. Calculating the sum, $a+b$, gives us $(\sqrt{3}+1)+(\sqrt{3}-1)=2\sqrt{3}$.

Now, we substitute these into our formula:

This simplifies to:

We know from our standard trigonometric values that $\cot (30^{\circ })=\sqrt{3}$. Therefore, the expression becomes:

This is the moment of magic! If $\tan \left(\frac{A-B}{2}\right)=1$, then the angle $\frac{A-B}{2}$ must be $45^{\circ }$. Consequently, $A-B=90^{\circ }$.

We have arrived at a simple system of linear equations: Equation 1: $A+B=120^{\circ }$ Equation 2: $A-B=90^{\circ }$

Solving this is a breeze. Adding the two equations gives $2A=210^{\circ }$, which means $A=105^{\circ }$. Subtracting the second from the first gives $2B=30^{\circ }$, so $B=15^{\circ }$.

The ratio $\angle A:\angle B$ is $105:15$, which simplifies beautifully to $7:1$.

This journey demonstrates that even complex-looking problems can be broken down into simple, logical steps. By mastering tools like Napier's Analogy, you transform from a student struggling with equations into an architect of geometric solutions.