Quadrilateral:
A quadrilateral is a polygon with four sides. It is a closed figure formed by connecting four line segments.

a) Two pairs of opposite angles:
Opposite angles in a quadrilateral are the angles that are opposite to each other when the lines intersect. In other words, they are not adjacent or next to each other.

Example of a quadrilateral with two pairs of opposite angles:

A
|\
| \
| \
| \
| \
| \
|_____\ B

In this example, angle A is opposite to angle B, and angle C is opposite to angle D.

b) Two pairs of adjacent angles:
Adjacent angles in a quadrilateral are the angles that share a common side. They are next to each other.

Example of a quadrilateral with two pairs of adjacent angles:

A______B
| |
| |
| |
| |
| |
D______C

In this example, angle A is adjacent to angle D, and angle B is adjacent to angle C.

c) Diagonals:
Diagonals in a quadrilateral are line segments that connect non-adjacent vertices. In other words, they are line segments that connect opposite corners of the quadrilateral.

Example of a quadrilateral with diagonals:

A______B
| |
| |
| x |
| |
|______|
C D

In this example, the diagonal AC connects vertex A to vertex C, and the diagonal BD connects vertex B to vertex D.

The diagonals of a quadrilateral have several properties. They bisect each other, meaning they divide each other into two equal parts. They also create four triangles within the quadrilateral. The sum of the measures of the opposite angles formed by the diagonals is always 180 degrees.

Overall, a quadrilateral is a versatile geometric shape with many properties and characteristics. Understanding its angles, sides, and diagonals helps us analyze and solve problems related to quadrilaterals in mathematics.