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Quadrangles

Quadrangles (also called quadrilaterals) are polygons with 4 edges, and 4 angles, or vertices,  at the corners.

Special quadrangles

Quadrangles in Geometry

There are several types of special polygons, with unique properties that depend on factors such as:

1. If their sides are parallel to each other or not - and if they are parallel, are both pairs of opposite edges parallel, or just one set

2. If their sides are all equal or if they just have two pairs of equal sides.

3. If the angles are right angles.

And more.

We have a section for each special type of polygon, describing and proving their properties, which are very common in high school geometry problems.

The sum of the angles in a quadrangle

One property that is common to all quadrangles, other than 4 sides and 4 vertices, is that the sum of the angles in a quadrangle is always 360°. The proof of this is simple.

Proof: Show that the sum of interior angles in a simple convex quadrangle is always 360°

Simple Convex Quadrangle

Let’s think about the strategy to do this. 360 is a familiar number - it is the measure of angles in a circle, but there are no circles here, so let’s rule that out.

360 is also 2x180, and that is also a familiar number – it is the sum of the angles in a triangle – so, if we can show that this quadrangle is composed of two triangles we will have the proof- and that’s all we need.

Proof: In any simple convex polygon, a line connecting 2 points on the perimeter of the polygon is entirely within the polygon, per the definition of convex polygons. So, let’s draw a line connecting two opposite corners of the quadrangle (such a line is a called a ‘diagonal’) – say from C to A:

Quadrangle with Diagonal

(1)    m∠A1 + m∠D+ m∠C1 = 180°              //sum of the interior angles in triangle ΔADC

(2)    m∠A2 + m∠B+ m∠C2 = 180°              //sum of the interior angles in triangle ΔABC

(3)    m∠A1 + m∠D+ m∠C1+m∠A2 + m∠B+ m∠C2 = 360°   //add both equations

(4)    m∠A1 + m∠A2 +m∠D+ m∠C1+ m∠C2 + m∠B= 360°   //re-arrange terms

(5)    m∠A1 + m∠A2 = m∠A                                      // angle addition postulate

(6)    m∠C1 + m∠C2 = m∠C                                       // angle addition postulate

(7) m∠A+m∠D+ m∠C + m∠B= 360°   ∎         //substitution property of equality

 

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Now that we've explained the basic concept of quadrangles in geometry, let's scroll down to work on specific geometry problems relating to this topic.

diagonals of kite

The Diagonals of a Kite are Perpendicular to Each Other

quadrilateral midpoints

Parallelogram Formed by Connecting the Midpoints of a Quadrilateral

Diagonals of a Rhombus are Perpendicular to Each Other

properties of parallelograms

Parallelograms: Consecutive Angles are Supplementary

parallelogram with diagonal

Parallelograms: The Two Pairs of Opposite Angles are Congruent

parallelogram with diagonal

Opposite Sides of a Parallelogram are Equal

properties of parallelograms

Proving that a Quadrilateral is a Parallelogram

trapezoid with angles

Trapezoids: Adjacent angles are Supplementary

Geometry shape: diamond with diagonals

Rhombus Diagonals Bisect the Angles

Geometry shape of a deltoid with diagonal

Deltoids: One of the Diagonals Bisects the Angles at its Endpoints

Square with diagonals

The Diagonals of Squares are Perpendicular to Each Other

Parallelogram with diagonals

The Diagonals of a Parallelogram Bisect Each Other

rectangle with diagonals

Diagonals of Rectangles are of Equal Length

The Sum of Interior Angles in a Simple Convex Quadrangle

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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