MATHEMATICS 7.6 CRITERIA FOR CONGRUENCE OF TRIANGLES We make use of triangular structures and patterns frequently in day-to-day life. So, it is rewarding to find out when two triangular shapes will be congruent. If you have two triangles drawn in your notebook and want to verify if they are congruent, you cannot everytime cut out one of them and use method of superposition. Instead, if we can judge congruency in terms of approrpriate measures, it would be quite useful. Let us try to do this. A Game Appu and Tippu play a game. Appu has drawn a triangle ABC (Fig 7.8) and has noted the length of each of its sides and measure of each of its angles. Tippu has not seen it. Appu challenges Tippu if he can draw a copy of his ΔABC based on bits of information thatAppu would give. Tippu attempts to draw a triangle congruent to ΔABC, using the information provided byAppu. The game starts. Carefully observe their conversation and their games. SSS GameFig 7.8 Triangle drawn by Appu : One side of ΔABC is 5.5 cm. Appu Tippu : With this information, I can draw any number of triangles (Fig 7.9) but they need not be copies of ΔABC. The triangle I draw may be obtuse-angled or right-angled or acute-angled. For example, here are a few. 5.5 cm 5.5 cm 5.5 cm (Obtuse-angled) (Right-angled) (Acute-angled) Fig 7.9 I have used some arbitrary lengths for other sides. This gives me many triangles with length of base 5.5 cm. So, giving only one side-length will not help me to produce a copy of ΔABC. Appu : Okay. I will give you the length of one more side. Take two sides of ΔABC to be of lengths 5.5 cm and 3.4 cm. Tippu : Even this will not be sufficient for the purpose. I can draw several triangles (Fig 7.10) with the given information which may not be copies of ΔABC. Here are a few to support my argument: Fig 7.10 One cannot draw an exact copy of your triangle, if only the lengths of two sides are given. MATHEMATICS EXAMPLE 3 In Fig 7.13, AD = CD and AB = CB. (i) State the three pairs of equal parts in ΔABD and ΔCBD. (ii) Is ΔABD ≅ΔCBD? Why or why not? (iii) Does BD bisect ∠ABC? Give reasons. SOLUTION (i) In ΔABD and ΔCBD, the three pairs of equal parts are as given below: AB = CB (Given) AD = CD (Given) and BD = BD (Common in both) (ii) From (i) above, ΔABD ≅ΔCBD (By SSS congruence rule)Fig 7.13 (iii) ∠ABD = ∠CBD (Corresponding parts of congruent triangles) So, BD bisects ∠ABC. CONGRUENCE OF TRIANGLES ABC is an isosceles triangle withAB = AC (Fig 7.17). A Take a trace-copy of ΔABC and also name it as ΔABC. (i) State the three pairs of equal parts in ΔABC and ΔACB. (ii) Is ΔABC ≅ΔACB? Why or why not? B (iii) Is ∠B = ∠C ? Why or why not? Fig 7.17 Appu and Tippu now turn to playing the game with a slight modification. SAS Game Appu : Let me now change the rules of the triangle-copying game. Tippu : Right, go ahead. Appu : You have already found that giving the length of only one side is useless. Tippu : Of course, yes. Appu : In that case, let me tell that in ΔABC, one side is 5.5 cm and one angle is 65°. Tippu : This again is not sufficient for the job. I can find many triangles satisfying your information, but are not copies of ΔABC. For example, I have given here some of them (Fig 7.18): Fig 7.18 MATHEMATICS 3. In Fig 7.24, measures of some parts of the triangles are indicated. By applying SAS congruence rule, state the pairs of congruent triangles, if any, in each case. In case of congruent triangles, write them in symbolic form. ASA Game Can you drawAppu’s triangle, if you know (i) only one of its angles? (ii) only two of its angles? (iii) two angles and any one side? (iv) two angles and the side included between them? Attempts to solve the above questions lead us to the following criterion: ASA 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 6 By applyingASAcongruence rule, it is to be established thatΔABC ≅ΔQRP and it is given that BC = RP. What additional information is needed to establish the congruence? CONGRUENCE OF TRIANGLES EXAMPLE 8 Given below are measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, using RHS congruence rule. In case of congruent triangles, write the result in symbolic form: ΔABC ΔPQR (i) ∠B = 90°, AC = 8 cm, AB = 3 cm ∠P = 90°, PR = 3 cm, QR = 8 cm (ii) ∠A = 90°, AC = 5 cm, BC = 9 cm ∠Q = 90°, PR = 8 cm, PQ = 5 cm SOLUTION(i) Here, ∠B = ∠P = 90º, hypotenuse, AC = hypotenuse, RQ (= 8 cm) and side AB = side RP ( = 3 cm) So, ΔABC ≅ΔRPQ (By RHS Congruence rule). [Fig 7.30(i)] (i) Fig 7.30 (ii) (ii) Here, ∠A = ∠Q (= 90°) and side AC = side PQ ( = 5 cm). But hypotenuse BC ≠hypotenuse PR [Fig 7.30(ii)] So, the triangles are not congruent. EXAMPLE 9 In Fig 7.31, DA ⊥AB, CB ⊥AB and AC = BD. State the three pairs of equal parts in ΔABC and ΔDAB. Which of the following statements is meaningful? (i) ΔABC ≅ΔBAD (ii) ΔABC ≅ΔABD Fig 7.31SOLUTION The three pairs of equal parts are: ∠ABC = ∠BAD (= 90°) AC = BD (Given) AB = BA (Common side) From the above, ΔABC ≅ΔBAD (By RHS congruence rule). So, statement (i) is true Statement (ii) is not meaningful, in the sense that the correspondence among the vertices is not satisfied. CONGRUENCE OF TRIANGLES We now turn to examples and problems based on the criteria seen so far. 1. Which congruence criterion do you use in the following? (a) Given: AC = DF AD AB = DE FCEBC = EF B So, ΔABC ≅ΔDEF (b) Given: ZX = RP R RQ = ZY ∠PRQ = ∠XZY PX So, ΔPQR ≅ΔXYZ (c) Given: ∠MLN = ∠FGH ∠NML = ∠GFHML = FG M So, ΔLMN ≅ΔGFH D (d) Given: EB = DB AE = BC ∠A = ∠C = 90° So, ΔABE ≅ΔCDB 2. You want to show that ΔART ≅ΔPEN, (a) If you have to use SSS criterion, then you need to show (i) AR = (ii) RT = (iii) (b) If it is given that ∠T = ∠N and you are to use SAS criterion, you need to have (i) RT = and (ii) (c) If it is given that AT = PN and you are to use ASA criterion, you need to have (i) ? (ii) ? MATHEMATICS 3. You have to show that ΔAMP ≅ΔAMQ. In the following proof, supply the missing reasons. 4. In ΔABC, ∠A = 30° , ∠B = 40° and ∠C = 110° Steps Reasons (i) PM = QM (i) ... (ii) ∠PMA = ∠QMA (ii) ... (iii) AM = AM (iii) ... (iv) ΔAMP ≅ ΔAMQ (iv) ... In ΔPQR, ∠P = 30° , ∠Q = 40° and ∠R = 110° A student says that ΔABC ≅ΔPQR by AAA congruence criterion. Is he justified? Why or why not? A5. In the figure, the two triangles are congruent. The corresponding parts are marked. We can N write ΔRAT ≅ ? 6. Complete the congruence statement: W Q RP T S ΔBCA ≅ ? ΔQRS ≅ ? 7. In a squared sheet, draw two triangles of equal areas such that (i) the triangles are congruent. (ii) the triangles are not congruent. What can you say about their perimeters? 8. If ΔABC and ΔPQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use?