Maharashtra Board Class 8 Maths part 2 Chapter 13 Congruence of triangles PDF Download

Congruence of Triangles

Let's Recall

Figures which exactly coincide with each other are called congruent figures.

The segments of equal lengths are congruent.

The angles of equal measures are congruent.

Teacher's Note

Congruent figures are exactly the same in shape and size. Like two identical tiles in your kitchen floor - they match perfectly.

Exam Trick

Remember: Congruent = Same shape and same size. If two figures look exactly the same when you place one on top of the other, they are congruent.

Points to Remember

Congruent figures look exactly the same.
Equal length segments are congruent.
Equal measure angles are congruent.
Congruence means perfect matching.

Let's Learn

Congruence of Triangles

Write answers to the following questions referring to the adjacent figure.

(i) Which is the angle opposite to the side DE?

(ii) Which is the side opposite to angle E?

(iii) Which angle is included by side DE and side DF?

(iv) Which side is included by angle E and angle F?

(v) State the angles adjacent to side DE.

Activity

Observe the adjacent figures. Copy triangle ABC on a tracing paper. Place it on triangle PQR such that point A coincides with point P, point B with point Q and point C with point R. You will find that both the triangles coincide exactly with each other, that is they are congruent.

In the activity, one way of placing triangle ABC on triangle PQR is given. But if we place point A on point Q, point B on point R and point C on point P, then two triangles will not coincide with each other. It means, the vertices must be matched in a specific way. The way of matching the vertices is denoted by one-to-one correspondence. Point A corresponds to point P is denoted as A ↔ P.

Here, two triangles are congruent in the correspondence A ↔ P, B ↔ Q, C ↔ R. When the two triangles are congruent in this way, we get six congruences, namely angle A ≅ angle P, angle B ≅ angle Q, angle C ≅ angle R, seg AB ≅ seg PQ, seg BC ≅ seg QR, seg CA ≅ seg RP.

Therefore, it is said that, triangle ABC and triangle PQR are congruent in the correspondence ABC ↔ PQR and written as triangle ABC ≅ triangle PQR.

Triangle ABC ≅ triangle PQR implies the correspondence A ↔ P, B ↔ Q, C ↔ R and the six congruences mentioned above. Therefore, while writing the congruence of two triangles, we have to take care that the order of vertices observes the one to one correspondence ascertaining congruence.

Teacher's Note

When we write triangle ABC ≅ triangle PQR, the order of letters matters. It tells us which vertex matches with which. Like matching a phone number - each digit in correct order gives the right number.

Exam Trick

When writing congruence, write vertices in the matching order. Triangle ABC ≅ triangle PQR means A matches P, B matches Q, and C matches R. Wrong order means wrong answer.

Points to Remember

Congruent triangles have the same shape and size.
One-to-one correspondence means matching vertices correctly.
The order of vertices in congruence statement is very important.
Congruent triangles have equal corresponding angles and sides.

Let's Discuss

Triangle ABC and triangle PQR are congruent. Their congruent parts are indicated by the identical marks.

Anil, Rehana and Surjit had written congruence of the triangles as follows.

Anil: triangle ABC ≅ triangle QPR

Rehana: triangle BAC ≅ triangle PQR

Surjit: triangle ABC ≅ triangle PQR

Which of the statements is correct and which is wrong? Discuss.

Solved Example

Ex. (1) In the adjacent figure, parts of triangles indicated by identical marks are congruent. (i) Identify the one to one correspondence of vertices in which the two triangles are congruent and write the congruence in two ways. (ii) State with reason, whether the statement, triangle XYZ ≅ triangle STU is right or wrong.

Solution: By observation, the triangles are congruent in the correspondence STU ↔ XZY hence

(i) Triangle STU ≅ triangle XZY is one way; triangle UST ≅ triangle YXZ is another way. Write the same congruence in some more different ways.

(ii) If the congruence is written as triangle XYZ ≅ triangle STU, it will mean side ST ≅ side XY, which is wrong.

Therefore statement triangle XYZ ≅ triangle STU is wrong.

(The writing triangle XYZ ≅ triangle STU includes some more mistakes. Students should find them out. Note that, to show that an answer is wrong it is sufficient to point out one mistake.)

Teacher's Note

Writing triangles in the wrong order can make correct congruences wrong. It is like writing your name in wrong order - it does not match you.

Exam Trick

Always match vertices in the correct order when writing congruence. Check the marks on the sides - they tell you which vertices match with which.

Points to Remember

Order of vertices matters in congruence statement.
Same triangles written in different order can be wrong.
Check correspondence before writing congruence.
Identical marks show which parts match.

Ex. (2)

In the given figure, the identical marks show the congruent parts in the pair of triangles. State the correspondence between the vertices of the triangles in which the two triangles are congruent.

Solution: In triangle ABD and triangle ACD, side AD is common. Every segment is congruent to itself. Therefore,

Correspondence: A ↔ A, B ↔ C, D ↔ D. Triangle ABD ≅ triangle ACD

Note: It is a convention to indicate a common side by the symbol '='.

Teacher's Note

A common side or angle is always congruent to itself. Like your face matching with your photograph - it is the same.

Exam Trick

When a side or angle is common to both triangles, always include it in your congruence. It is one of the three parts needed to prove congruence.

Points to Remember

A common side is congruent to itself.
A common angle is congruent to itself.
Always use common parts in your proof.
Common parts help prove triangle congruence.

Let's Learn

To show that a pair of triangles is congruent, it is not necessary to show that all six corresponding parts of the two triangles are congruent. If three specific parts of one triangle are respectively congruent with the three corresponding parts of the other, then the remaining three corresponding parts are also congruent with each other. It means, the specific three parts ascertain the test of congruence.

We have learnt to construct triangles. The three specific parts of a triangle which define a unique triangle decide a test of congruence. Let us verify this.

(1) Two Sides and the Included Angle: SAS Test

Draw triangle ABC and triangle LMN such that two pairs of their sides and the angles included by them are congruent.

Draw triangle ABC and triangle LMN, l(AB) = l(LM), l(BC) = l(MN), m∠ABC = m∠LMN

Copy triangle ABC on a tracing paper. Place the paper on triangle LMN in such a way that point A coincides with point L, side AB overlaps side LM, B overlaps M and side BC overlaps side MN. You will notice that triangle ABC ≅ triangle LMN.

Teacher's Note

SAS test means Side-Angle-Side. The angle must be between the two sides. Like making a sandwich - the filling must be between the two bread slices.

Exam Trick

Remember: In SAS test, the angle must be INCLUDED between the two sides. The angle sits between them like a person between two chairs.

Points to Remember

SAS = Side-Angle-Side test.
The angle must be between the two sides.
Two equal sides and included angle prove congruence.
Order: Side, then Angle, then Side.

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