How would I geometrically prove this situation of a rectangle and a triangle? I.e., how is theta used twice?

Question

Here is a situation of Euclidian geometry, a proof of some sorts.

I am quite unsure how to explain what was done. This is from an AP Physics B (Part 1) prep book, and this was the first step in the answering process, which made too many logical steps. The prompt was this:

The prompt.

What I am asking of is theta; I noted that it was used twice, and I too am aware that some geometric proof is involved in relating the two triangles.

I am sorry for the vagueness of this situation, and I hope the information is sufficient enough. How would I prove that theta of the top triangle is theta on the side? (Perhaps this should be moved to the Physics exchange if too vague in jargon.)

Answer 1

So I'm labelling that 2nd angle as $\beta$. So we want to show that $\beta = \theta$

So looking at the triangle with the angle $\alpha$ and $\beta$, we know the sum of the angles of a triangle are 180

So

$\alpha+\theta+90 = 180$

$\theta = 90-\alpha$

We also know that since angles on a straight line add to 180

$\alpha+90+\beta = 180$

So $\beta = 90-\alpha$

Therefore $\beta = \theta$ so that's why they're labeled the same.

Answer 2

Soo, it sounds like you just need to be convinced that both angles marked as $\theta$ are indeed equal? This is because the two triangles they are parts of are similar:

• They clearly both have a right angle.
• The unmarked angle in the upper triangle is $180 - (90+\theta) = 90-\theta$, since the triangle's angles sum to $180$. The unmarked angle in the lower triangle makes $180$ degrees together with the original $\theta$ (from the upper triangle) and a right angel; so that is also $90-\theta$.

And so the third angle of the two triangles must be equal as well.