The total angle must be $ 90 ^{\circ} $ degrees however as we sum up the each angle , the equation be dishold

Question

I've drawn the above diagram as same as the diagram of the book.

The leftmost and the right most arrows make $~90^{\circ}~$

Currently I can't get how the below 2 angles are obtained.

$$ \theta_{1}:= \theta_{} + 45 ^{\circ} =\text{angle between left M and H} $$

$$ \theta_{2}:= \theta_{} - 45 ^{\circ} =\text{angle between right M and H} $$

$$ \theta_{1} + \theta_{2} = 90 ^{\circ} $$

must be held but actually

$$ \theta_{1} + \theta_{2} =2 \theta_{} \neq 90 ^{\circ} $$

I've may made some mistake.

I think we may can assume $~\theta \ll1~$

Answer 1

The above circled part shows your error. It should be $45-\theta$ and not the other way round.

Now continue and you get $\theta+45$ and $45-\theta$. Now can you complete the answer yourself?

Answer 2

It looks like:

$\theta_1 + \theta_2 = 2\times\theta $ and definitely does not equal $90$.

The sum is not supposed to be equal to $90$ as $2\times\theta$ is not equal to $90$.

On the other hand: $\theta_1 - \theta_2 = 90$

Answer 3

Angle between M and H is ($45-\theta$) not ($\theta-45$). Angle between M and H ,left one,is ($45+\theta$) or ($\theta+45$). So the sum of two angles is:

$\theta+45+45-\theta=90$