AI Tutor
If you tear the three corners off any paper triangle and line them up edge-to-edge, they always form a perfectly flat line of 180∘180^\circ.
It works whether the triangle is tall and skinny, wide and flat, or perfectly equilateral — but how do we prove it is true for every possible triangle in the universe?
To build an airtight mathematical proof, what secret geometry tools do we need to unlock?

The Two Secret Tools

First, a straight angle is formed by rays pointing in opposite directions along a single line, measuring exactly 180∘180^\circ.
Second, when two parallel lines are crossed by a third line called a transversal, the alternate interior angles (angles sitting on opposite sides of the transversal between the parallel lines) are always congruent (equal in measure).
Around 300 BCE in ancient Alexandria, the mathematician Euclid realized he could connect these parallel line rules to triangles by adding one extra line to the sketch.
How can adding an extra line to a triangle instantly reveal its angle sum?

The Auxiliary Line

An auxiliary line is an extra line or segment added to a geometric diagram to make relationships easier to see and prove.
Given any △ABC\triangle ABC, we construct a single line ℓ\ell that passes through top vertex BB and runs perfectly parallel to the opposite base AC‾\overline{AC}.