NCERT MATHS CLASS 7 गणित कक्षा 7
Congruence of Triangles त्रिभुजों की सर्वांगसमता
LINK TO SOLUTIONS OF ALL QUESTIONS IS GIVEN BELOW
सभी प्रश्नों के समाधान का लिंक नीचे दिया गया है
CLASS 7 CHAPTER 7 Congruence
of Triangles
1. Congruent objects are exact copies of one another.
2. The method of superposition examines the congruence of plane figures.
3. Two plane figures, say, F1 and F2 are congruent if the trace-copy of F1 fits exactly on that of F2 . We write this as F1 ≅ F2 .
4. Two line segments, say, AB and CD , are congruent if they have equal lengths. We write this as AB≅ CD . However, it is common to write it as AB = CD .
5. Two angles, say, ∠ABC and ∠PQR, are congruent if their measures are equal. We write this as ∠ABC ≅ ∠PQR or as m∠ABC = m∠PQR. However, in practice, it is common to write it as ∠ABC = ∠PQR.
6. SSS Congruence of two triangles: Under a given correspondence, two triangles are congruent if the three sides of the one are equal to the three corresponding sides of the other.
7. SAS Congruence of two triangles: Under a given correspondence, two triangles are congruent if two sides and the angle included between them in one of the triangles are equal to the corresponding sides and the angle included between them of the other triangle.
8. ASA Congruence of two triangles: Under a given correspondence, two triangles are congruent if two angles and the side included between them in one of the triangles are equal to the corresponding angles and the side included between them of the other triangle.
9. RHS Congruence of two right-angled triangles: Under a given correspondence, two right-angled triangles are congruent if the hypotenuse and a leg of one of the triangles are equal to the hypotenuse and the corresponding leg of the other triangle.
10. There is no such thing as AAA Congruence of two triangles: Two triangles with equal corresponding angles need not be congruent. In such a correspondence, one of them can be an enlarged copy of the other. (They would be congruent only if they are exact copies of one another).
ERCISE 7.1
CONGRUENCE OF ANGLES AND TRIANGLES
कोणों और त्रिभुजों की
सर्वांगसमता
1. Complete the following statements: (a) Two line segments are congruent if ___________. (b) Among two congruent angles, one has a measure of 70°; the measure of the other angle is ___________. (c) When we write ∠A = ∠B, we actually mean ___________.
2. Give any two real-life examples for congruent shapes.
3. If ∆ABC ≅ ∆FED under the correspondence ABC ↔ FED, write all the corresponding congruent parts of the triangles.
4. If ∆DEF ≅ ∆BCA, write the part(s) of ∆BCA that correspond to
(i) ∠E (ii) EF(Line) (iii) ∠F (iv) DF(Line)
1. निम्नलिखित कथनों को पूरा कीजिए: (a) दो रेखाखंड सर्वांगसम होते हैं यदि ___________। (बी) दो सर्वांगसम कोणों में से एक का माप 70° है; दूसरे कोण का माप ___________ है। (सी) जब हम ∠A = ∠B लिखते हैं, तो वास्तव में हमारा मतलब ___________ होता है।
2. सर्वांगसम आकृतियों के लिए कोई दो वास्तविक जीवन उदाहरण दीजिए।
3. यदि अनुरूपता ABC ↔FED के अंतर्गत ∆ABC≅ ∆FED है तो त्रिभुजों के सभी संगत सर्वांगसम भागों को लिखिए।
4. यदि ∆DEF ≅∆BCA है, तो ∆BCA का वह भाग लिखिए जो के संगत है
(i) E (ii) EF (लाइन) (iii) ∠F (iv) DF (लाइन)
EXERCISE
7.2
CRITERIA FOR CONGRUENCE OF TRIANGLES
and CONGRUENCE AMONG
RIGHT-ANGLED TRIANGLES
1. Which congruence criterion do you use in the following?
(a) Given:AC = DF AB = DE BC = EF So, ∆ABC ≅ ∆DEF
(b) Given: ZX = RP RQ = ZY ∠PRQ = ∠XZY So, ∆PQR ≅ ∆XYZ
(c) Given: ∠MLN = ∠FGH ∠NML = ∠GFH ML = FG So, ∆LMN ≅ ∆GFH
(d) Given: EB = DB AE = BC ∠A = ∠C = 90° So, ∆ABE ≅ ∆CDB
2. You want to show that ∆ART ≅ ∆PEN,
(a) If you have to use SSS criterion, then you need to show (i) AR = (ii) RT = (iii) AT =
(b) If it is given that ∠T = ∠N and you are to use SAS criterion, you need to have (i) RT = and (ii) PN =
(c) If it is given that AT = PN and you are to use ASA criterion, you need to have (i) ? (ii) ?
3. You have to show that ∆AMP ≅ ∆AMQ.
In the following proof, supply the missing reasons. Steps Reasons
(i) PM = QM (i) ...
(ii) ∠PMA = ∠QMA (ii) ...
(iii) AM = AM (iii) ...
(iv) ∆AMP ≅ ∆AMQ (iv) ...
4. In ∆ABC, ∠A = 30° , ∠B = 40° and ∠C = 110° In ∆PQR, ∠P = 30° , ∠Q = 40° and ∠R = 110° A student says that ∆ABC ≅ ∆PQR by AAA congruence criterion. Is he justified? Why or why not?
5. In the figure, the two triangles are congruent. The corresponding parts are marked. We can write ∆RAT ≅ ?
6. Complete the congruence statement: ∆BCA ≅ ? ∆QRS ≅ ?
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. Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent.
9. If ∆ABC and ∆PQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use?
10. Explain, why ∆ABC ≅ ∆FED.
1. निम्नलिखित में आप किस सर्वांगसमता मानदंड का उपयोग करते हैं?
(ए) दिया गया है: एसी = डीएफ एबी = डीई बीसी = ईएफ तो, ∆ABC DEF
(बी) दिया गया है: ZX = RP RQ = ZY PRQ = ∠XZY तो, ∆PQR ∆XYZ
(सी) दिया गया है: MLN = ∠FGH ∠NML = ∠GFH ML = FG तो, LMN GFH
(d) दिया गया है: EB = DB AE = BC A = C = 90° तो, ABE CDB
2. आप दिखाना चाहते हैं कि ART PEN,
(ए) यदि आपको एसएसएस मानदंड का उपयोग करना है, तो आपको दिखाना होगा (i) एआर = (ii) आरटी = (iii) एटी =
(बी) यदि यह दिया गया है कि ∠T = ∠N और आपको एसएएस मानदंड का उपयोग करना है, तो आपके पास (i) RT = और (ii) PN = होना चाहिए।
(सी) यदि यह दिया गया है कि एटी = पीएन और आपको एएसए मानदंड का उपयोग करना है, तो आपके पास (i) होना चाहिए? (ii)?
3. आपको दिखाना है कि AMP AMQ.
निम्नलिखित प्रमाण में छूटे हुए कारणों की पूर्ति कीजिए। कदम कारण
(i) पीएम = क्यूएम (i) ...
(ii) ∠PMA = ∠QMA (ii) ...
(iii) AM = AM (iii) ...
(iv) AMP AMQ (iv) ...
4. ABC में, A = 30°, ∠B = 40° और ∠C = 110° PQR में, P = 30°, Q = 40° और ∠R = 110° एक छात्र कहता है कि ∆ABC AAA सर्वांगसमता मानदंड द्वारा PQR। क्या वह जायज है? क्यों या क्यों नहीं?
5. आकृति में, दोनों त्रिभुज सर्वांगसम हैं। संबंधित भागों को चिह्नित किया गया है। हम RAT लिख सकते हैं?
6. सर्वांगसमता कथन को पूरा करें: BCA ? क्यूआरएस ?
7. एक वर्गाकार शीट में समान क्षेत्रफल वाले दो त्रिभुज इस प्रकार खींचिए कि (i) त्रिभुज सर्वांगसम हों। (ii) त्रिभुज सर्वांगसम नहीं हैं। आप उनकी परिधि के बारे में क्या कह सकते हैं?
8. दो त्रिभुजों का एक मोटा रेखाचित्र इस प्रकार खींचिए कि उनमें सर्वांगसम भागों के पाँच जोड़े हों लेकिन फिर भी त्रिभुज सर्वांगसम न हों।
9. यदि ABC और PQR को सर्वांगसम होना है, तो संगत भागों के एक अतिरिक्त युग्म के नाम लिखिए। आपने किस मानदंड का उपयोग किया?
10. समझाइए, क्यों ABC
FED।
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