Proofs In Plane Geometry

Properties Of Circles

1. tan. ⊥ rad.

2. rt. ∠ in semicircle

3. ∠s in same seg.

4. ∠ at centre = 2 ∠ at circ.

5. ∠s in opp. seg. (a+c=b+d=180°)

6. tan. from ext. pt.

Congruent & Similar Triangles

$$\begin{array}{|c|c|} \hline \text{Congruent triangles} & \text{Similar triangles}\\\hline \text{SSS, SAS, AAS, RHS} & \text{SSS, SAS, AAA}\\\hline \end{array}$$

Midpoint Theorem

If $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively, then $DE$ // $BC$ and $DE = \frac{1}{2}BC$.

Tangent-Chord Theorem (Alternate Segment Theorem)

If $DE$ is a tangent to the circle at $B$, then $\angle CAB = \angle CBE$ and $\angle ACB = \angle ABD$.