Two Quadrilaterals With the Same Side Lengths Are Always Congruent

Two Quadrilaterals With the Same Side Lengths Are Always Congruent

03 November 2020


Read time: 5 minutes


Introduction

Sizes and shapes are the backbones of geometry. Ane of the most encountered shapes in geometry is polygons. The Greek word ‘Polygon’ consists of
Poly
meaning ‘many’ and
gon
meaning ‘angle.’

Polygons are two-dimensional shapes equanimous of straight lines. They are said to have a ‘closed shape’ as all the lines are connected. In this article, we volition discuss the concept of similarity in Polygons.


Like Polygons – PDF

If you lot ever want to read it once more as many times equally you want, hither is a downloadable PDF to explore more.



Similar Polygons

Start, let us go clear with what ‘similar’ ways. Two things are called like when they both take a lot of the same properties only however may not exist identical. The aforementioned tin can exist said well-nigh polygons.


Congruent polygons


As you might take studied, Coinciding shapes are the shapes that are an exact match.
Congruent polygons
accept the aforementioned size, and they are a perfect lucifer as all respective parts are coinciding or equal.


Similar polygons definition

On the other hand, In
Similar polygons, the respective angles are congruent, but the corresponding sides are proportional. And so, similar polygons have the same shape, whereas their sizes are unlike. At that place would be sure compatible ratios in similar polygons.


Properties of similar polygons

There are ii crucial backdrop of like polygons:

  • The respective angles are equal/congruent. (Both interior and exterior angles are the same)
  • The ratio of the respective sides is the aforementioned for all sides. Hence, the perimeters are different.
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two similar polygons(triangles), whose corresponding angles are equal

The above image shows two similar polygons(triangles), ABC, and A’B’C’. We tin run into that respective angles are equal.

\[<A=<A’, <B=<B’,<c=<C’\]

The respective sides have the aforementioned ratios.

\[\frac{AB}{A’B’}=\frac{BC}{B’C’}=\frac{CA}{C’A’}\]



Like Quadrilaterals

Quadrilaterals are polygons that take four sides. The sum of the interior angles of a quadrilateral is 360 degrees. Ii quadrilaterals are similar quadrilaterals when the three corresponding angles are the same( the fourth angles automatically become the same as the interior angle sum is 360 degrees), and two adjacent sides have equal ratios.


Are all squares similar?

Let us discuss the similarity of squares. According to the similarity of quadrilaterals, the respective angles of similar quadrilaterals should exist equal. Nosotros know that all angles are ninety degrees in the square, so all the corresponding angles of whatsoever 2 squares will be the aforementioned.

Similarity of squares, all sides of a square are equal

All sides of a square are equal. If let’s say, square1 has a side length equal to ‘a’ and square2 has a side length equal to ‘b’, so all the corresponding sides’ ratios will be the same and equivalent to a/b.

Hence, all squares are similar squares.


Are all rhombuses like?

In a Rhombus, all the sides are equal. So, only similar squares, rhombuses satisfy the condition of the ratio of corresponding sides being equal.

In Rhombus all the sides are equal

In a Rhombus, the reverse sides are parallel, and hence the opposite angles are equal. Only the value of those angles can exist anything. So, it can very much happen that two rhombuses have different angles. Hence, all rhombuses are not like.



Similar Rectangles

Two rectangles are similar when the corresponding adjacent sides have the same ratio. Nosotros do not demand to check the angles as all angles in a rectangle are 90 degrees.

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Two rectangles are similar when the corresponding adjacent sides have same ratio.

In the higher up paradigm, the ratios of the adjacent side are . Hence, these are like rectangles.


Are all rectangles similar?

No, all rectangles are not like rectangles. The ratio of the respective adjacent sides may be dissimilar. For example, let’s accept a 1 past 2 rectangle and take another rectangle with dimensions i past iv. Here the ratios will not be equal.

\[\frac{1}{1}\ne\frac{4}{2}\]


Congruent rectangles

Two rectangles are called coinciding rectangles if the corresponding adjacent sides are equal. It means they should have the same size. The expanse and perimeter of the coinciding rectangles volition too be the same.



Summary

Similarity and congruency are some important concepts of geometry. A solid understanding of these topics helps in building a good foundation in geometry. This article discussed the concepts of similarity in polygons looking at some specific cases of similar quadrilaterals like similar squares, similar rectangles, and similar rhombuses.



Frequently Asked Questions (FAQs)

What are like polygons?

Two polygons are similar when the corresponding angles are equal/coinciding, and the corresponding sides are in the same proportion.

If two rectangles take the same perimeter, are they coinciding?

No, rectangles are not always coinciding when they have the same perimeter. The ratio of lengths of respective sides may exist different fifty-fifty when the perimeter is the same. For instance, a rectangle of 5 by 4 and another rectangle of 6 by iii has the aforementioned perimeter(equal to 18), but the corresponding sides’ ratios are dissimilar \(\frac{v}{6}\ne\frac{4}{3}\).

Two Quadrilaterals With the Same Side Lengths Are Always Congruent

Source: https://www.cuemath.com/learn/similar-polygons/