If a transversal intersects two parallel lines, then prove that each pair of alternate interior angles is equal.

If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

If two lines intersect each other, then prove that the vertically opposite angles are equal.

If two circles intersect at two points, then prove that their centres lie on the perpendicular bisector of the common chord.

Two parallel lines l and m are intersected by a transversal p (see the given figure). Show that the quadrilateral formed by the bisector of interior angles is a rectangle.

I and m are two parallel lines intersected by another pair of parallel lines p and show that triangle ABC = triangle CDA

If each pair of opposite sides of a quadrilateral is equal, then prove that it is a parallelogram.

In the adjacent figure, a line n falls on lines 1 and m such that the sum of the interior angles 1 and 2 is less than 180°, then what can you say about lines 1 and m.

If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.

overset(harr)("AB ") and overset(harr)("DC ") are two parallel lines and a transversal l, intersects overset(harr)("AB ") at P and overset(harr)("DC ") at R. Prove that the bisectors of the interior angles form a rectangle.

If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.