Two line segments ab and cd are congruent. if ab = 6 cm, what is the length of cd?

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Any two line segments are congruent if and only if their lengths are equal.

In the above figures, we see the length of the line segment AB is equal to the length of the line segment CD. Hence, two line segments AB and CD are congruent.

Also, if two line segments are congruent, we sometimes just say that the line segments are equal.

Here, AB ≅ CD is same as AB = CD

∵AB = CD = 5 cm

Example: Find the given line segments are congruent or not?

We know any two line segments are congruent if and only if their lengths are equal.

Here, we see the lengths of the two line segments are not equal. Hence, they are not congruent with each other.

∵AB ≠CD

Example: Two line segments AB and PQ are congruent. If AB = 5 cm, then what is the length of PQ?

The line segment AB and PQ are congruent.

∴ AB = PQ

Given that AB = 5 cm ⇒ PQ = 5 cm

Congruence of Angles

Two angles are congruent if and only if their measures are equal.

In the above figure, when we superimpose ∠PQR on ∠ABC, we see two arms of ∠PQR are completely superimpose on ∠ABC and their
measures are also the same i.e.,40°

So, we can say that

∠ABC ≅∠PQR

∠ABC = ∠PQR

Also, if the angles are congruent, their measures are the same.

Here, we know ∠ABC ≅∠PQR

∴∠ABC = ∠PQR

∵∠ABC = ∠PQR = 40°

Example: Find the given angles are congruent or not?

In the above figures when we superimpose ∠PQR on ∠ABC, we see two arms of ∠PQR are not completely superimposed on ∠ABC and their measures are also different.

Hence, ∠ABC is not congruent with ∠PQR

Or ∠ABC ≠∠PQR

Example: Two angles ∠ABC and ∠XYZ are congruent. If ∠ABC = 75°, then what is the measure of ∠XYZ?

We have,

∠ABC ≅∠XYZ

We know if the angles are congruent, their measures are the same.

∴∠ABC = ∠XYZ

Given that ∠ABC = 65°

⇒∠XYZ = 65°

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Two triangles are congruent if they are exact copies of each other and cover each other exactly.

In the above figure when we place ∆PQR on ∆ABC we see both triangles are exactly the same shape and size also they cover each other exactly. So, they are congruent to each other. We express this as ∆ABC ≅ ∆PQR.

The alphabetical order of naming triangles is very important as they represent the corresponding parts of the triangle which means
when we place ∆PQR on ∆ABC we see,

The vertex A coincides with vertex P, vertex B coincides with vertex Q and vertex C coincides with vertex R. Here correspondence is represented using ⟷.

The side AB coincides with side PQ, side AC coincides with side PR and side BC coincides with side PQ.

Here, vertex and sides are coinciding with each other so, we can say that their angles are also matching with each other.

m∠BAC = m∠QPR, m∠ACB = m∠PRQ and m∠CBA = m∠RQP

Thus, in these two congruent triangles, we have:

Corresponding vertices: A ⟷P, B ⟷Q, C ⟷R.

Corresponding sides: AB⟷PQ, BC ⟷QR, AC ⟷PR.

Corresponding angles: ∠A ⟷∠P, ∠B ⟷∠Q, ∠C ⟷∠R.

Example: If ∆PQR ≅ ∆XYZ write the part(s) of ∆PQR that corresponds to (i) PQ (ii) ∠A (iii) YZ

Here, ∆PQR ≅ ∆XYZ

∆PQR ⟷ ∆XYZ

(i)PQ ⟷ XY (ii) ∠A ⟷∠P (iii) YZ ⟷QR

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In the previous section, we learnt that two triangles are congruent if all six pairs of corresponding parts of the two triangles are congruent, or in other words, we can say that all the three pairs of corresponding sides must be congruent and all the three pairs of corresponding angles must be congruent.

In this section, we learn how to prove two triangles are congruent using only three pairs of corresponding parts. So, there are four conditions for the congruency of two triangles. Let us see them one by on

(i) SSS (Side – Side – Side) Congruence criterion

If under a given condition, the three sides of one triangle are congruent to three sides of another triangle, then two triangles are congruent.

[Note: Here we don’t need to know about angles.]

Example:

In ∆ABC and ∆PQR, we have

AB = PQ, BC = QR and AC = PR
Hence, ∆ABC and ∆PQR are congruent by side–side–side congruence criterion.

∴∆PQR ≅ ∆XYZ

Example: Check whether two triangles ABC and triangles LMN are congruent.

Solution: In ∆ ABC and ∆LMN, we have

BC = MN = 3 cm (Given)

⇒AB = LN = 4.5 cm (Given)

⇒AC = LM = 4.8 cm (Given)

Therefore, ∆ ABC ≅ ∆LMN (By SSS-criterion of congruence)

Example: In Fig, PS = RS and PQ = RQ.

(i) State the three pairs of equal parts in ΔPQS and ΔRQS.

(ii) Is ΔPQS≅ΔRQS? Why?

(i)In ΔPQS and ΔRQS, the three pairs of equal parts are as given below:

PQ = RQ... (Given)

PS = RS ... (Given)

and

QS = QS ... (Common in both)

Hence, ΔPQS≅ΔRQS

(By SSS congruence rule)

(ii) SAS (Side –Angle– Side) Congruence criterion

If under a correspondence, two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle, then two triangles are congruent.

Example:

In ∆ABC and ∆PQR, we have

AB = PQ = 5 cm (Given)

∠BAC = ∠QPR = 60° (Given)

AC = PR = 4 cm (Given)

Therefore, ∆ABC ≅ ∆PQR (By SAS-criterion of congruence)

Example: In Fig, PQ = PR and PS is the bisector of ∠QPR.

(i) State three pairs of equal parts in triangles PSQ and PSR.

(ii) Is ΔPSQ ≅ΔPSR? Give reasons.

(iii) Is ∠Q = ∠R?

(i) The three pairs of equal parts are as follows:

PQ = PR (Given)

∠QPS = ∠RPS (PS bisects ∠QPR) and PS = PS (common)

(ii) Yes, ΔPSQ ≅ΔPSR (By SAS congruence rule)

(iii) ∠Q = ∠R (Corresponding parts of congruent triangles)

(iii) ASA (Angle – Side –Angle) Congruence criterion

If under a correspondence, two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, then the triangles are congruent.

Example:

In ∆ABC and ∆PQR, we have

∠ABC = ∠PQR = 40°... (Given)

BC = QR = 6 cm ... (Given)

∠BCA = ∠QRP = 60°... (Given)

Therefore, ∆ ABC ≅∆ PQR (By ASA-criterion of congruence)

Example: In the following figure can you say that ∆LON ≅∆MOP

In the two triangles ∆LON and ∆MOP

∠LNO = ∠MPO = 75° ... (Given)

∆LON =∆MOP = 20° ... (Vertically opposite angles)

In ∆LON,

∠OLN + ∠LNO + ∠NOL = 180°... (Vertically opposite angles)

∠OLN = 180°−(∠LNO + ∠NOL)

∠OLN= 180°−(75° + 20°)

∠OLN= 180°−( 95°)

∠OLN= 85°

In ∆MOP,
∠OMP + ∠MPO + ∠POM = 180°... (Vertically opposite angles)

∠OMP = 180°−(∠MPO + ∠POM)

∠OMP = 180°−(75° + 20°)
∠OMP = 180°−( 95°)

∠OMP = 85°

Thus, we have ∠OLN = ∠OMP, LN = PM and ∠LNO = ∠MPO

Now, side LN is between ∠OLN and ∠LNO and side PM is between ∠MPO and ∠OMP.

So, by ASA rule ∆LON ≅∆MOP

(iv) RHS (Right angle – Hypotenuse – Side) Congruence criterion

If under a correspondence, hypotenuse and one side of the right angle triangle are respectively equal to the hypotenuse and one side of the other right angle triangle, then the triangles are congruent.

Example:

In ∆ABC and ∆PQR, we see

∠ABC = ∠PQR = 90°(Given)

AC = PR = 5 cm (Given)

AB = PQ = 3 cm (Given)

Therefore, ∆ABC ≅ ∆PQR (By RHS-criterion of congruence)

Example: In the given diagram check whether ∆ADC and∆CBA hold the property of RHS congruence.

In ∆ADC, ∠ADC = 90°

In ∆CBA, ∠CBA = 90°

AC = 6 cm (hypotenuse is same for both ∆ADC and ∆CBA)

AB = BC = 3 cm

So, ∆ADC and ∆CBA hold the property of the RHS congruence rule,

∆ADC ≅ ∆CBA

Prove that the diagonals of the parallelogram bisect each other.

In the parallelogram ABCD, diagonals AC and BD intersect at O.

It is required to prove that AO = OC and BO = OD

In ∆AOD and ∆COB, we have

∠OAD = ∠OCD (Alternate angles as AD || BC and AC is the transversal)

Similarly ∠ODA = ∠OBC

AD = BC (Opposite sides of a parallelogram)

∴ ∆AOD≅ ∆COB (ASA congruency)

∴AO = OC and BO = OD (Corresponding parts of congruent triangles)

Hence, the diagonals of the parallelogram bisect each other is proved.

Prove that the bisector of the vertical angle of an isosceles triangle bisects the base at a right angle.

Given: AB = AC and AD is the bisector of ∠A

To prove: ∠ADB = ∠ADC = 90oand BD = DC

Proof: In ∆ADB and ∆ADC, we have

AB = AC ... (given)

AD = AD ... (Common)

∴ ∆ADB≅∆ADC (SAS congruency)

So, BD = DC and ∠ADB = ∠ADC ....(i)

But, ∠ADB + ∠ADC = 180°

2∠ADB = 180°

∠ADB =

° = 90°

∴ ∠ADB = ∠ADC = 90°

(v) AAA (Angle-Angle-Angle) Rule

In ΔABC and ΔPQR, we have

∠BAC = ∠QPR ..... (Given)

∠ACB= ∠PRQ ..... (Given)

∠CBA= ∠RQP ..... (Given)

Hence, ΔABC and ΔPQR are similar but not congruent because their sizes are different.

Hence, we do not use AAA congruence criterion.

Example: In ΔABC, ∠A= 30°, ∠B = 50°∠C = 100°

In ΔPQR, ∠P = 30°, ∠Q = 50°∠R = 100°

Here ΔABC ≅ΔPQR by AAA congruence criterion. Is it true?

Here, we have

∠A = ∠P = 30°

∠B = ∠Q = 50°

∠C = ∠R = 100°

Here, all the three angles of both triangles are equal but sides may not be equal. So, triangles may not be congruent. Hence, the given statement is false.