##### Shmoop Study Guides

(Sunnyvale - United States)Shmoop wants to make you a better lover (of literature, history, poetry and writing). See many sides to the argument. Find your writing groove. Understand how lit and history are relevant today. We want to show your brain a good time. Our mission: To make ...

## Quadrilaterals

**Description:**

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Geometry

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

**Instructional Component Type(s):**- Curriculum: Unit

- Contributed By: Winpossible

### Getting Started - Quadrilaterals

**Description:**

In this mini-lesson you'll learn; taking help of given examples with solution, the properties of a quadrilateral. A quadrilateral is a four-sided polygon with four angles i.e. anything with four sides that are connected so that a closed figure is obtained and that is a polygon, is an example of a quadrilateral. There are many types of quadrilaterals. The five most common types of quadrilaterals are the parallelogram, the rectangle, the square, the trapezoid, and the rhombus.

You can further understand taking the case of four points

*A*,

*B*,

*C*, and

*D*in a plane such that no three of them are collinear and where the line segments

*AB*,

*BC*,

*CD*, and

*DA*intersect at the endpoints only, then figure formed by four such line segments is a quadrilateral. Line segment drawn from one vertex of a quadrilateral to the opposite vertex is called diagonal of the quadrilateral. E.g.

*AC*and

*BD*are the diagonals of quadrilateral

*ABCD*. You will also learn in this section that there are many different kinds of quadrilaterals, but all have several things in common. For example, all of them have four sides, are coplanar, have two diagonals, and the sum of their four interior angles equals 360 degrees.

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Geometry

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

**Instructional Component Type(s):**- Asset: Video/Presentation/Slides

In this mini-lesson you'll learn; taking help of given examples with solution, the properties of a quadrilateral. A quadrilateral is a four-sided polygon with four angles i.e. anything with four sides that are connected so that a closed figure is obtained and that is a polygon, is an example of a quadrilateral. There are many types of quadrilaterals. The five most common types of quadrilaterals are the parallelogram, the rectangle, the square, the trapezoid, and the rhombus.

You can further understand taking the case of four points

*A*,

*B*,

*C*, and

*D*in a plane such that no three of them are collinear and where the line segments

*AB*,

*BC*,

*CD*, and

*DA*intersect at the endpoints only, then figure formed by four such line segments is a quadrilateral. Line segment drawn from one vertex of a quadrilateral to the opposite vertex is called diagonal of the quadrilateral. E.g.

*AC*and

*BD*are the diagonals of quadrilateral

*ABCD*. You will also learn in this section that there are many different kinds of quadrilaterals, but all have several things in common. For example, all of them have four sides, are coplanar, have two diagonals, and the sum of their four interior angles equals 360 degrees.

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

### Properties of Quadrilaterals

**Description:**

Here you'll learn the important properties of quadrilaterals and how to apply the correct property to each different type of quadrilateral. The interactive learning of lessons and practice questions with solution, is presented by instructor with the help of video of their voice and own handwriting.

The three properties that we are going to look at are - the number of sides, interior angles (the angles inside) and the length of the sides. A quadrilateral is a polygon with four sides and it has four vertices. You'll have explanation; with the help of examples including solution, of the adjacent sides, opposite sides, adjacent angles or consecutive angles and opposite angles of a quadrilateral.

Two sides of a quadrilateral, which have a common vertex, are called adjacent sides E.g. in quadrilateral *ABCD*, *AB* and *BC*, *BC* and *CD*, *CD* and *DA*, *DA* and *AB* are the pairs of adjacent sides. Two sides of a quadrilateral are opposite, if they do not have a common endpoint i.e. *AB* and *DC*, *BC* and *AD* are two pairs of opposite sides.

Two angles of a quadrilateral having rays on the same line are known as adjacent angles or consecutive angles i.e. *A* and *B*, *B* and *C*, *C* and *D*, *D* and *A* are the pairs of adjacent angles. Two angles of a quadrilateral that do not have rays on the same line are known as opposite angles E.g. *A* and *C*, *B* and *D*, are pairs of opposite angles.

In quadrilateral *ABCD*, diagonal *AC* divides quadrilateral into two triangles Δ*ABC* and Δ*ADC*. We know from earlier learnings that sum of the interior angles of a triangle is 180°; therefore, sum of the interior angles of a quadrilateral is 180° + 180° = 360°. If the measurements of three angles of a quadrilateral are known, then the missing angle can be calculated. For example, in a quadrilateral *ABCD*, *ABC* is a right angle, *BCD* = 70° and *BAD* = 100°, then measure of *ADC* equals to 100°.

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Geometry

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

**Instructional Component Type(s):**- Curriculum: Lesson Plan

Here you'll learn the important properties of quadrilaterals and how to apply the correct property to each different type of quadrilateral. The interactive learning of lessons and practice questions with solution, is presented by instructor with the help of video of their voice and own handwriting.

The three properties that we are going to look at are - the number of sides, interior angles (the angles inside) and the length of the sides. A quadrilateral is a polygon with four sides and it has four vertices. You'll have explanation; with the help of examples including solution, of the adjacent sides, opposite sides, adjacent angles or consecutive angles and opposite angles of a quadrilateral.

Two sides of a quadrilateral, which have a common vertex, are called adjacent sides E.g. in quadrilateral *ABCD*, *AB* and *BC*, *BC* and *CD*, *CD* and *DA*, *DA* and *AB* are the pairs of adjacent sides. Two sides of a quadrilateral are opposite, if they do not have a common endpoint i.e. *AB* and *DC*, *BC* and *AD* are two pairs of opposite sides.

Two angles of a quadrilateral having rays on the same line are known as adjacent angles or consecutive angles i.e. *A* and *B*, *B* and *C*, *C* and *D*, *D* and *A* are the pairs of adjacent angles. Two angles of a quadrilateral that do not have rays on the same line are known as opposite angles E.g. *A* and *C*, *B* and *D*, are pairs of opposite angles.

In quadrilateral *ABCD*, diagonal *AC* divides quadrilateral into two triangles Δ*ABC* and Δ*ADC*. We know from earlier learnings that sum of the interior angles of a triangle is 180°; therefore, sum of the interior angles of a quadrilateral is 180° + 180° = 360°. If the measurements of three angles of a quadrilateral are known, then the missing angle can be calculated. For example, in a quadrilateral *ABCD*, *ABC* is a right angle, *BCD* = 70° and *BAD* = 100°, then measure of *ADC* equals to 100°.

### Special Quadrilaterals

**Description:**

This mini-lesson is on special types of quadrilaterals. So far the focus of learning was on properties of quadrilaterals. Now you'll explore, with the help of some examples and solution, how these properties and knowledge will be used for special quadrilaterals- a parallelogram, rectangle, square, rhombus, trapezoid, an isosceles trapezoid and kites.

A parallelogram is a quadrilateral, if both pairs of opposite sides are equal and parallel to each other. Also in case of parallelogram, opposite sides and opposite angles are congruent and diagonals bisect each other.

A rectangle is a parallelogram with four right angles. Opposite sides of a rectangle are parallel and equal. The diagonals of a rectangle are congruent. A square is rectangle with four congruent sides. Opposite sides of a square are parallel and all sides are equal in length and all angles are equal to 90°.

A rhombus is a parallelogram with four congruent sides. Opposite sides are parallel and opposite angles are equal. The diagonals of a rhombus are perpendicular and bisect each other.

A trapezoid is a quadrilateral with exactly one pair of parallel sides. An isosceles trapezoid is a quadrilateral where two opposite sides are parallel, the two other sides are of equal length. Also the diagonals are of equal length and base angles are congruent.

A kite is a quadrilateral with two pairs of adjacent sides is equal in length. One pair of opposite angles is equal in measure. The two diagonals of a kite are perpendicular and bisect each other.

Here are the important properties to remember:

- All quadrilaterals have 4 sides.
- A square has got 4 sides of equal length and 4 right angles (right angle = 90 degrees).
- A Rhombus has got 4 sides of equal length and opposite sides are parallel and angles are equal.
- The rectangle (oblong) contains 4 right angles (an angle of 90˚). It has got 2 pairs of equal length sides.
- A parallelogram is a rectangle that has been pushed over. Opposite sides are the same length and they are parallel.
- A trapezium has got one pair of parallel sides.
- A kite as got two pairs of sides next to each other that have equal length.

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Geometry

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

**Instructional Component Type(s):**- Curriculum: Lesson Plan

This mini-lesson is on special types of quadrilaterals. So far the focus of learning was on properties of quadrilaterals. Now you'll explore, with the help of some examples and solution, how these properties and knowledge will be used for special quadrilaterals- a parallelogram, rectangle, square, rhombus, trapezoid, an isosceles trapezoid and kites.

A parallelogram is a quadrilateral, if both pairs of opposite sides are equal and parallel to each other. Also in case of parallelogram, opposite sides and opposite angles are congruent and diagonals bisect each other.

A rectangle is a parallelogram with four right angles. Opposite sides of a rectangle are parallel and equal. The diagonals of a rectangle are congruent. A square is rectangle with four congruent sides. Opposite sides of a square are parallel and all sides are equal in length and all angles are equal to 90°.

A rhombus is a parallelogram with four congruent sides. Opposite sides are parallel and opposite angles are equal. The diagonals of a rhombus are perpendicular and bisect each other.

A trapezoid is a quadrilateral with exactly one pair of parallel sides. An isosceles trapezoid is a quadrilateral where two opposite sides are parallel, the two other sides are of equal length. Also the diagonals are of equal length and base angles are congruent.

A kite is a quadrilateral with two pairs of adjacent sides is equal in length. One pair of opposite angles is equal in measure. The two diagonals of a kite are perpendicular and bisect each other.

Here are the important properties to remember:

- All quadrilaterals have 4 sides.
- A square has got 4 sides of equal length and 4 right angles (right angle = 90 degrees).
- A Rhombus has got 4 sides of equal length and opposite sides are parallel and angles are equal.
- The rectangle (oblong) contains 4 right angles (an angle of 90˚). It has got 2 pairs of equal length sides.
- A parallelogram is a rectangle that has been pushed over. Opposite sides are the same length and they are parallel.
- A trapezium has got one pair of parallel sides.
- A kite as got two pairs of sides next to each other that have equal length.