In the previous lesson we learnt about triangles. Now we know what triangles can be and how they differ from each other as well as what common features all triangles have. In this lesson, we will learn what a quadrangle is. Let's begin our lesson with the definition of a quadrangle.

**A quadrangle**is a quadrangular geometric figure formed by a closed broken line consisting of four segments. Points in a quadrangle are called vertices, segments are called sides or faces of a quadrangle.

The quadrangle includes the following simple figures:

- Four points (
,**A**,**B**,**C**);**D** - Four segments (
,**AB**,**BC**,**CD**);**DA** - Four corners (
,**α**,**β**,**γ**);**θ** - A closed broken line (
).**ABCD**

## Vertices of the quadrangle

The vertices of the quadrangle are defined with capital Latin letters. The quadrangle is denoted by a sequence of vertices. In the image we have the quadrangle * ABCD*.

## Sides of the quadrangle

The sides of the quadrangle (or so-called faces or edges) are denoted by lowercase Latin letters. In the image, we have designated the sides of the quadrangle with letters * a*,

*,*

**b***,*

**c***. Also, triangle sides can be defined as segments, for example,*

**d***,*

**a = AB***,*

**b = BC***,*

**c = CD****d = DA**. Sides belonging to the same vertex are called adjacent parties. In the image for the vertex A we have adjacent sides

*and*

**AD***.*

**AB**## Quadrangle angles

As we know, the angles are defined with lowercase Greek letters. In the image, the angles of the quadrangle are denoted by * α*,

*,*

**β***,*

**γ***, where*

**θ***,*

**α = DAB***,*

**β = ABC***,*

**γ = BCD***. Each corner of the quadrangle has an opposing angle, for example, in the figure the angle*

**θ = CDA***has an opposing angle*

**α***, and for*

**γ***the opposite angle*

**β***.*

**θ**## Quadrangle diagonals

**The diagonal of a quadrangle**is a segment joining the opposite vertices of a quadrangle.

Overall, the quadrangle has two diagonals. In the image, segments * AC* and

*are the diagonals of the quadrangle*

**BD***.*

**ABCD**If quandrangle diagonals intersect at a right angle, it is called **orthodiagonal**.

## Quadrangle bisectors

**The bisector of a quadrilateral**is a ray dividing the angle of the vertex into two equal angles.

During the lesson about triangles, we described how a bisector of a vertex can be found with the help of a protractor.

## Quadrangle middle lines

**he middle line of the quadrangle**is the segment connecting the opposite sides through their midpoints.

In order to draw the middle line, you need to split all sides of the quadrangle into two equal segments and to find the midpoints. The midpoints of the opposite sides must be joined by segments that will be quadrangle middle lines.

## Classification of quadrangles

### Convex quadrangles

**A convex quadrangle**is a quadrangle that has all the corners convex (from 0° to 180°, not including these values). The sum of all the angles of a convex quadrangle should always be 360°

### Non convex or concave quadrangles

**A concave quadrangle**is a quadrangle having one concave angle (180° to 360°, not including these values). The sum of all the angles of a concave quadrangle should always be 360°.

### Self-intersecting quadrangles

**A self-intersecting quadrangle**is the one having intersecting opposite sides.

A quadrangle that is visually self-intersecting is similar to two triangles with the sides that lie on two intersecting lines, and their intersection point is a common vertex for triangles.

## Types of quadrangles and their features

### Parallelogram

**A parallelogram**is a quadrangle having opposite sides parallel and equal. The parallelogram has equal opposite angles.

### Rectangle

**A rectangle**is a parallelogram with equal angles. A rectangle is a quadrangle having opposite sides parallel and equal. All corners of the rectangle should be 90°.

### Rhombus

**A rhombus**is a parallelogram with equal sides. A rhombus is an orthogonal quadrangle having all sides equal and opposite sides parallel.

### Square

**A square** is a rectangle with equal sides. A square is a convex quadrangle having all sides and angles equal. All angles of the square should be 90°.

### Trapezoid

**A trapezoid**is a quadrangle that has two parallel opposite sides. These sides are called the base of the trapezoid, the other sides are called the hips of the trapezoid.

The trapezoid with equal hips is called an isosceles trapezoid. The angles at the bases are equal in an isosceles trapezium. The image to the right represents an** isosceles trapezoid**.

### Deltoid

**A deltoid**is an orthogonal quadrangle with equal adjacent sides for opposite angles.

### Antiparallelogram

**An antiparallelogram**(

**counterparallelogram**) is a self-intersecting quadrangle with equal but not parallel opposite sides.

If we draw a convex geometric figure through antiparallelogram vertices, we will get an isosceles trapezoid.

## How to write a circle into a quadrangle?

A circle tangent to all four quadrangle sides is called the inscribed circle of a quadrangle. A circle can be inscribed in a quadrangle only under the condition that **the sum of the opposite sides is equal**. If it is not so, it is impossible to inscribe a circle in a quadrangle.

**The center of the inscribed circle of a quadrangle** is the point of intersection of all the medians of the quadrangle. If we measure the distance along the perpendicular from this point to any base, we will get the radius of the inscribed circle.

## How to describe a quadrangle with a circle?

A circle passing through all vertices of a quadrangle is called the circumscribed circle. If the **sum of the opposite angles is 180°**, this quadrangle might be described by a circle. If it cannot be satisfied, the quadrangle cannot be described by a circle.

**The center of the circumscribed circle of a quadrilateral** is the point of intersection of perpendiculars coming from side midpoints.

We have already drawn the circumscribed circle for a triangle. The algorithm to describe the circle for a quadrangle is similar. The image clearly represents how from the middle of the sides we held perpendiculars and found the intersection point. With the help of the caliper, we set the radius from the center to any vertex and draw the circumscribed circle of the quadrangle with a circular motion.

Find orthogonal quadrangles.

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