PROOFS OF SOME BASIC THEOREMS

CONTENT

• (i) Riders including angles of parallel lines
• (ii) Angles in a polygon
• Congruent triangles
• Properties of Parallelogram
• (v) Intercept theorem

Angles of Parallel Lines

Recall: Basic geometrical facts are called theorems. The first is the sum of the angles of a triangle is 180⁰. Many other theorems depend on it. For this reason they are often called Riders. (Since they ride on theorem 1)

If two parallel lines are intersected by a transversal;

(i) the alternate angles are equal.

(ii) the corresponding angles are equal.

(iii) the interior angles on the same side of the transversal are supplementary viz:

(a)

Transversal

\(a = a\) (Alternate angles)

(b)

\(b = b\) (Corresponding angles)

(c)

Co-interior/allied angles

\(c + d = 180^o\) (Supplementary angles)

Other angles formed are: vertically opposite angles, they are equal.

Angle at a point (\(360^o\)) and on a straight line (\(180^o\))

\(a = a\) (Vertically opposite angles)

Examples:

1.