Core Ideas in Plain Terms

The Triangle Angle-Sum Theorem says the three angles of any triangle add to $180^\circ$; a quadrilateral adds to $360^\circ$ — both come from the polygon angle-sum formula $(n-2) \cdot 180^\circ$. So a missing angle is just the total minus the ones you know. When a line crosses two parallel lines, it makes matching (equal) angles — the key to "angle chasing."

In a right triangle, the two short sides (legs) and the longest side (hypotenuse) satisfy the Pythagorean theorem: $a^2 + b^2 = c^2$. A few side-triples come up constantly — $3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$ — so recognizing them saves time.

Two figures are similar if one is a scaled copy of the other — all angles equal, all sides in the same ratio. If you scale lengths by $k$, areas scale by $k^2$ and volumes by $k^3$. That is why doubling a shape's size quadruples its area.

An angle drawn from the circle's edge sees an arc at half the angle drawn from the center. Two consequences worth knowing cold: every inscribed angle in a semicircle is $90^\circ$, and inscribed angles on the same arc are equal.

The distance between two points is $\sqrt{(\Delta x)^2 + (\Delta y)^2}$ — the hypotenuse of the right triangle their coordinates make. From $(0,0)$ to $(3,4)$: $\sqrt{9+16} = 5$. The midpoint just averages the coordinates.

When a figure gives you nothing to compute with, add an auxiliary line: a radius to a tangent point, an altitude, a diagonal. Each new line creates triangles — and triangles bring the Pythagorean theorem, similarity, and area formulas into play.