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Triangles

Congruent Triangles

Triangles are the most primary shapes we learn. As closed figures with three-sides, triangles are of different types depending on their sides and angles. The common variants are equilateral, isosceles, scalene etc. What are congruent triangles then, in this chapter we shall learn about the same.

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Congruent Triangles

We all know that a triangle has three angles, three sides and three vertices. Depending on similarities in the measurement of sides, triangles are classified as equilateral, isosceles and scalene. The comparison done in this case is between the sides and angles of the same triangle. When we compare two different triangles we follow a different set of rules.

Two similar figures are called congruent figures. These figures are a photocopy of each other. You must have noticed two bangles of the same size, and shape, these are said to be congruent with each other. When an object is exactly similar to the other, then both are said to be congruent with each other.

Every congruent object, when placed over its other counterpart, seems like the same figure. Similarly, congruent triangles are those triangles which are the exact replica of each other in terms of measurement of sides and angles. Let’s take two triangles If Δ XYZ  and Δ LMN.

Both are equal in sides and angles. that is, side XY = LM, YZ = MN and ZX= NL. When these two triangles are put over each other, ∠X covers ∠L, ∠Y covers ∠M and ∠N covers ∠Z. Both these triangles are said to be congruent to each other and are written as  Δ XYZ ≅ Δ LMN.

It must, however, be noted that Δ XYZ ≅ Δ LMN but Δ ZYX is not congruent to Δ LMN. This means that it is not necessary that the triangle be congruent to each other if the sides are inverted the other way round.

Congruent triangles

This makes it clear that the correct representation of sides and vertices is necessary, to show that two triangles are congruent to each other.

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Properties of Congruent Triangles

1. Sides and Angles

Before understanding the necessary criterion for congruence it is essential that you understand how many equal sides and angles make a congruent pair. For this, it is necessary that you do the following activity. Draw two triangles, with one of the sides of both triangles measuring 5 cm. Do these triangles look congruent?

Congruent triangles

The answer is no! Now redraw these triangles with one of the angles being 45° and one side 5cm. What do you notice? These two aren’t congruent as well!

Now re-do the same activity with two equal sides and one equal angle, forming the same two sides. What do you notice? The resulting triangles seem similar. This brings us to a conclusion that for two triangles to be congruent, they should have two equal sides and one equal angle comprising the same sides. The figure below will make things clear:

Congruent triangles

This is the very first criterion of congruence.

2. SAS Congruence Rule

SAS stands for Side-Angle-Side. A triangle is said to be congruent to each other if two sides and the included angle of one triangle is equal to the sides and included angle of the other triangle. This axiom is an accepted truth and does not need any proofs to support the criterion.

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3. ASA Congruence Rule

ASA stands for Angle Side Angle congruence. Two triangles are said to be congruent to each other if two angles and the included side of one triangle is equal to the two angles and the included side of the other triangle. When we have to prove that two triangles are equal, through this criterion we look at the following aspect of two triangles:

In triangles ABC and PQR, we know that, ∠ B = ∠Q, ∠C = ∠R and BC=QR. Now we need to prove that ΔABC ≅ ΔPQR. We prove the same by considering three cases:

Case 1:  Let AB= PQ, this means that
∠ B = ∠ Q
BC= QR
So by SAS rule, ΔABC ≅ ΔPQR

Case 2: If possible let AB > PQ,
Now we take a point O on AB such that OB= PQ
Now consider triangles OBC and PQR

In ΔOBC and ΔPQR,
OB=PQ,
∠B =∠Q
and BC= QR
So by the SAS axiom, we conclude that ΔOBC ≅ ΔPQR

Since the triangles are congruent to each other, their related parts shall also be equal, so ∠ACB =∠PRQ and ∠ACB =∠OCB, which is possible only if O coincides with A, or if BA= QP. So, ΔABC ≅ ΔPQR

Case 3: 

If AB < PQ, we choose a point X on PQ so that XQ = AB. Using the case 2 theory for the ΔPQR, we can prove that ΔABC ≅ ΔPQR.

4. AAS Congruence

AAS stands for Angle-Angle Side congruence. Two triangles are congruent to each other if any of the two pairs of angles and one pair of corresponding sides are equal to each other. The basis of this theory is the Angle sum property of triangles.

According to the angle sum property, the sum of three angles in a triangle is 180°. So if two triangles are equal, automatically the third side is also equal, hence making triangles perfectly congruent.

5. SSS Congruence

SSS congruence means Side Side Side congruence. In two triangles, is all the three sides of one triangle are equal to the three sides of the other triangle, then both triangles are said to be congruent with each other.

6. RHS congruence

RHS stands for Right angle Hypotenuse Side congruence. In two right-angled triangle, if the hypotenuse and one side of a triangle are equal to the hypotenuse and one side of the other triangle, then both the triangles are congruent to each other. From the above discussion, we can now understand the basic properties of congruence in triangles.

Solved Examples for You

Question 1: In a rectangle PQRS, QS is diagonal. Then ΔPQS ________________ ΔRSQ.

  1. is similar to
  2. is not equal
  3. is congruent to
  4. is not congruent to

Answer : In a rectangle PQRS, QS is diagonal.
We know that in a rectangle opposite sides are equal.
∴PQ = SR
Also, = 900
Ans QS is the hypotenuse for both right-angled triangles PQS and RSQ. Thus, according to R.H.S condition for congruence, ΔPQS ≅ ΔRSQ.

Question 2: What are the tests of congruence in a triangle?

Answer: The various tests of congruence in a triangle are: SAS, SSS, ASA, AAS, and HL. These tests tell us about the various combinations of congruent angles and/or sides that help in determining whether the two triangles are congruent.

Question 3: Can we say that AAA is a valid congruence theorem?

Answer: AAA is not a valid theorem of congruence. This is because; it is not necessary for triangles that have 3 pairs of congruent angles to have the same size.

Question 4: Explain the SSS postulate?

Answer: Side Side Side (SSS) postulate states that if three sides of one triangle happen to be congruent to three sides belonging to another triangle, then these two triangles are said to be congruent.

Question 5: Explain the SAS similarity theorem?

Answer: SAS is certainly a valid theorem of congruence. This theorem is one of the ways of proving that the triangles are congruent. This particular theorem states that if one triangle’s angle is congruent to another triangle’s corresponding angle, while the lengths of the sides are in proportion including these angles, then the triangles are said to be similar.

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