Geometric shapes are look-alikes of familiar-looking forms of objects. Accordingly, whatever shape you encounter in your daily life is called a ‘geometric shape’.
In the world of geometry, congruence is an imperative topic as when one object is equal to the other object. For instance, identical-looking twins stand congruent to each other as they are look-alikes.
For identifying congruent shapes, you’d require only basic knowledge of geometry.
So, without wasting much time, let us understand the congruence of two forms, in the detailed explanations given in the table below.
|Congruency||Identical two figures, if they are congruent.|
|Incongruent||Non-identical looking models.|
Describe Congruent Figures
In geometry, two figures or objects with similar shapes or sizes are mathematically known as congruent. These figures look like the mirror image of the other. For a better understanding, let us have a look at a couple of examples –
- You have an artwork which is perfect for your bedroom’s wall – neither too narrow nor too wide. You like it so much that you put a similar size one in your living room. A friend of yours, who loves mathematics, might be of a different opinion. To him, both the artwork stands congruent to each other!
With two congruent figures, they stand precisely of similar shape and size. Their outlook might differ owing to one being shifted or rotated in a particular manner, but they are still a similar shape. And all the sides of each one are of identical length, likewise the matching sides of the other.
How to Recognise Congruent Shapes?
Congruency stands true for two ‘congruent’ figures. Let us understand the same through a couple of examples.
Example#1 – Two Congruent Shapes
Jane works at her father’s interior designing agency. Her father asks her to find out a wheel’s identical copy for replacement, having a 12″ radius. What should be the radius of the exact copy that Jane has to find-out?
The answer is simple! If some object stands identical to some other object, they are congruent. And owing to their congruence, all the dimensions remain similar.
Example#2 – Two Non-Congruent Shapes
Misha packs some gift boxes together for delivery. She is almost done with the packing but has room for just one item. The left-over space is enough for any rectangular box measuring 5″ wide by 7″ long by 3″ high. She is left with three small-size packs measuring 4″ wide by 4″ long by 4″ tall, 3″ wide by 7″ long by 2″ high, and 4″ wide by 6″ long by 3″ tall, respectively. Which of these boxes might perfectly fit inside the remaining space?
To fit something correctly, it requires to be of a similar size. If it stands of the equivalent size, then it stands ‘congruent’, with all its critical dimensions remaining identical. But no boxes have similar dimensions to fit inside left-over space, owing to their non-congruent shapes.
You might have studied with the detailed knowledge of a two-dimensional triangle having three vertices, three sides and three angles. More than two triangles stand congruent, with their related triangle-sides. In simpler terms, the triangles with similar form and measurements are Congruent.
The term ‘Congruency’ describes a couple of objects with a similar size and shape, with the congruence symbol being ≅. The usage of the short term CPCT is prevalent, showing the similarity between the Congruent Triangles’ concerned parts.
Congruent Sides of a Triangle
Let us talk-over the congruence concerning two triangles, namely ΔABCΔABC and ΔPQRΔPQR. These two triangles stand of similar shape and size, with the same ‘congruent’.
The detailed explanation lies like –
- AA falls on PP
- BB falls on QQ
- CC falls on RR
- ABAB falls on PQPQ
- BCBC falls on QRQR
- ACAC falls on PRPR
Therefore, the whole equation indicates the corresponding parts of congruent triangles being equal.
|Corresponding Angles||∠A∠A and ∠P∠P|
∠B∠B and ∠Q∠Q
∠C∠C and ∠R∠R
|Corresponding Vertices||AA and PP|
BB and QQ
CC and RR
|Corresponding Sides||ABAB and PQPQ|
BCBC and QRQR
ACAC and PRPR
While finding out the same, always be mindful of the fact that it happens to be inappropriate to write ΔBAC≅ΔPQR, ΔBAC≅ΔPQR as AA agrees to PP, BB agrees to QQ and CC corresponds to RR.
We can also represent the same in a mathematical formula with the use of the ‘congruent’ symbol – ΔABC≅ΔPQR, ΔABC≅ΔPQR.
Triangle Congruence and its four properties
The congruence in a Triangle consists of the methods used in proving whether two triangles are congruent or not. The four criteria for testing triangle congruence consist of:
- Side – Angle – Side (SAS)
- Angle – Angle – Side (AAS)
- Side – Side – Side (SSS)
- Angle – Side – Angle (ASA)
It is essential for having the knowledge in marking the different angles and sides with a specific sign, showing their Congruency, before discussing the postulates of congruence in detail.
While extending our discussion further, you will come across the four criteria, mentioned below, having the sides of similar measurement.
SAS stands for Side-Angle-Side (mathematical terminology), a rule proving if the triangle sets are congruent or not. In this situation, two triangles remain congruent. Its one included angle and two-sides in a given triangle should remain identical to the angle and corresponding two-sides of another triangle.
While calculating the same, you should keep in mind the angle formed by two given triangle sides to become congruent. For example, if DE = XY, DF = XZ, and ∠ YXZ = ∠ EDF, then; the triangles DEF and XYZ are congruent (△DEF ≅△ XYZ).
AAS stands for Angle-Angle-Side (mathematical terminology), a rule stating two sides of the congruent triangles included out of three. And the corresponding two angles of the triangles remain equal. For example, being ∠ BAC = ∠ QPR, ∠ ACB = ∠ RQP, and length of AB = QR, then the triangles ABC and PQR remain congruent (△ABC ≅△ PQR).
SSS stands for Side-Side-Side (mathematical terminology) stating two consecutive triangles remain congruent if the lengths of three-sides remain equal. For example, the DEF and XYZ triangles are congruent (△DEF ≅△ XYZ), provided their length DE = XZ, DF = YX, and EF = YZ.
ASA stands for Angle-Side-Angle (mathematical terminology) It states two triangles being congruent, with their matching two angles, along with the one equal side. For example, the triangles ABC and PQR are congruent (△ABC ≅△ PQR), if length ∠ BAC = ∠ PRQ, ∠ ACB = ∠ PQR.
Solved Examples of Triangle Congruence:
- Two triangles BCD and RST are such that; BC = 4.5 cm, CD = 6.1 cm, BD = 6 cm, RS = 6.1 cm, ST = 6 cm and RT = 4.5 cm. Find out whether these triangles are congruent.
Given: BC = RT = 4.5 cm
BC = RS = 6.1 cm and
BD = ST = 6 cm
Hence, ∆BCD ≅ ∆RST (SSS)
- Give a description of the congruence type found in two triangles; ∆ BCD, BC = 7 cm, CD = 4 cm, ∠C = 60° and ∆ EFG, EF = 4 cm, FG = 7 cm, ∠F = 60°
BC = FG = 7 cm,
CD = EF = 5 cm and
∠C =∠F = 50°
Hence, ∆BCD ≅ ∆EFG (SAS)
Congruent Objects – Day-to-day Examples
We encounter many congruent shapes in our day-to-day life; it is very easy to identify them. Right from some food packets, stationary objects or daily utility items, congruence has become a part of our daily life. For instance, a simple one is a biscuit packet having similar shapes and sizes of cookies. Thus, we can say all the cookies are congruent. Or a deck of cards. All the cards lie similar to each other in one packet! Following is a list of congruent examples, you may find in your routine:
- A packet of cigars.
- Pages of any book.
- Ten fingers in our hands.
- A box of match-sticks.
- A box of color pencils.
So with this article, we have learned about the real meaning of congruence. For instance, with two congruent shapes, all their critical dimensions stand similar too. Do you still want to get your doubts cleared related to Basic and Advanced Geometry? And want to clarify the same with friendly explanations on congruence and not textual notes? Our Cuemath’s expert Math tutors are your solution; they hold well-communicative online classes and assist students in building confidence. So, get on the right track and turn your child into a Mathematics expert by availing of the trial class for free today!