AI Tutor
How did ancient astronomers calculate the distance to the Moon without leaving the ground?
They used a simple geometric superpower: if two right triangles share the same sharp angle, their side ratios are locked into exact matching values, no matter how big or small the triangles are.
What guarantees that every triangle with that angle grows in exact lockstep?

The AA Similarity Guarantee

Recall that the interior angles of any planar triangle sum to 180∘180^\circ.
In any right triangle with an acute reference angle ΞΈ\theta, the remaining acute angle is automatically fixed at 90βˆ˜βˆ’ΞΈ90^\circ - \theta.
By the Angle-Angle (AA) similarity criterion, any two triangles with two congruent corresponding angles are similar.
If two right triangles are similar, what happens when we divide one side by another?

Why Ratios Belong to the Angle

Let small triangle β–³ABC\triangle ABC have sides aa, bb, and hypotenuse cc. Any larger similar triangle β–³Aβ€²Bβ€²Cβ€²\triangle A'B'C' has side lengths scaled by a positive constant kk: kak a, kbk b, and kck c.
Look at the ratio of the vertical leg to the hypotenuse in the larger triangle:
kakc=ac\frac{k a}{k c} = \frac{a}{c}