7.1 class 9

Transcript Theorem 7.1 (ASA Congruence Rule) :- Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle. Given :- ABC and DEF such that B = E & C = F and BC = EF To Prove :- ABC DEF Proof:- We will prove by considering the following cases :- Case 1: Let AB = DE In ABC and DEF AB = DE B = E BC = EF ABC DEF Case 2: AB > DE Construction :- Take a point P on AB such that PB = DE In PBC and DEF PB = DE B = E BC = EF PBC DEF PCB = DFE But ACB = DFE Thus, ACB = PCB This is possible only if P is coincides with A AB = DE By Case 1 ABC DEF Case 3: If AB < DE If AB < DE, then by choosing a point M on DE such that AB = ME and repeating the argument in Case (2). We get ABC DEF Show More

Triangles Class 9 NCERT Book: If you are looking for the best books of Class 9 Maths then NCERT Books can be a great choice to begin your preparation. NCERT Class 9th Maths Chapter 7 Books will give you authentic information and you can rely on them during your preparation. Try practicing the previous papers and sample questions attached in the NCERT Books of Class 9th Maths Chapter 7 Triangles to solve the questions in your exam easily. Class 9 Maths Chapter 7 Triangles NCERT Book PDF Download To ace in your exam preparation, you can refer to the 9th Class NCERT Solutions prevailing in NCERT e-Book. NCERT Books for Class 9 Maths Triangles will have illustrative problems and solutions. Students can understand the concepts written in NCERT 9th Class Maths Textbooks for Ch 7 Triangles well as all of them are written in a comprehensive matter. NCERT Textbooks of Class 9th Maths Chapter 7 Triangles are free of cost and you can access them through the quick links available. • • FAQs on NCERT Books for 9th Class Maths Triangles PDF 1. Why one should read NCERT Books for Class 9 Maths Chapter 7? Students can self evaluate the knowledge gap with the help of NCERT Books for 9th Class Maths Chapter 7 and also gain more knowledge on the concepts of Maths. 2. Where can I download NCERT Books for Class 9 Maths Chapter 7 Triangles PDF free? You can download NCERT Books for 9th Class Maths Chapter 7 Triangles PDF free of cost on our page. 3. How to download Class 9 Maths Triangles NCERT ...

AD = BC ………………………………………………… (iii) From above three equation both the triangle satisfies “AAS” congruency criterion So, ΔAOD ≅ ΔBOC AO and BO will be equal as they are corresponding parts of congruent triangles(CPCT). So, AO = BO Hence, CD bisects AB at O. Question4. l and m are two parallel lines intersected by another pair of parallel lines p and q (see Fig. 7.19). Show that ∆ ABC ≅ ∆ CDA. Solution: Given that l and m are two parallel lines p and q are another pair of parallel lines Considering two triangles ΔABC and ΔCDA Where, ∠ BCA = ∠DAC ………….(iii) From above three equation both the triangle satisfies “ASA” congruency criterion So, ΔABC ≅ ΔCDA Question5. Line l is the bisector of an angle ∠ A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A (see Fig. 7.20). Show that: (i) ∆ APB ≅ ∆ AQB (ii) BP = BQ or B is equidistant from the arms of ∠ A. Solution: Given that, Line l is the bisector of an angle ∠ A and B BP and BQ are perpendiculars from angle A. Considering two triangles ΔAPB and ΔAQB Where, ∠ APB = ∠ AQB ……… (iii) From above three equation both the triangle satisfies “AAS” congruency criterion So, ΔAPB≅ ΔAQB. (ii) Also we can say BP and BQ are equal as they are corresponding parts of congruent triangles(CPCT). So, BP = BQ Question6. In Fig. 7.21, AC = AE, AB = AD and ∠ BAD = ∠ EAC. Show that BC = DE. Solution: Given that AC = AE, AB = AD And ∠BAD = ∠EAC As given that ∠BAD = ∠EAC Adding ∠DAC on both the sides We get, ∠BAD + ∠DAC = ∠EAC ...

NCERT Exemplar Class 9 Maths Solutions Triangles Exercise 7.1: Multiple Choice Questions (MCQs) Question 1: Which of the following is not a criterion for congruence of triangles? (a) SAS (b) ASA (c) SSA (d) SSS Thinking Process For triangle to be congruent it is very important that the equal angles are included between the pairs of equal sides. So, SAS congruence rule holds but not ASS or SSA rule. Solution: (c) We know that, two triangles are congruent, if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle. Also, criterion for congruence of triangles are SAS (Side-Angle-Side), ASA (Angle-Side- Angle), SSS (Side-Side-Side) and RHS (right angle-hypotenuse-side). So, SSA is not a criterion for congruence of triangles. Question 2: If AB = QR, BC = PR and CA = PQ, then (a) ΔABC ≅ ΔQRP (b) ΔCBA ≅ ΔPRQ (c) ΔBAC ≅ ΔRQP (d) ΔPQR ≅ ΔBCA Solution: (b) We know that, if ΔRST is congruent to ΔUVW i.e., ΔRST = ΔUVW, then sides of ΔRST fall on corresponding equal sides of ΔUVW and angles of ΔRST fall on corresponding equal angles of ΔUVW. Here, given AB = QR, BC = PR and CA = PQ, which shows that AB covers QR, BC covers PR and CA covers PQ i.e., A correspond to Q, B correspond to R and C correspond to P. or A↔Q,B↔R,C↔P Under this correspondence, ΔABC ≅ ΔQRP, so option (a) is incorrect, or ΔBAC ≅ ΔRQP, so option (c) is incorrect, or ΔCBA ≅ ΔPRQ, so option (b) is correct, or ΔBCA ≅ ΔRPQ, so option (d) is incorrect. Question 3: In ΔABC...

NCERT Solutions for Class 9 Maths Chapter 7 Triangles Ex 7.1 are part of • • • • • Board CBSE Textbook NCERT Class Class 9 Subject Maths Chapter Chapter 7 Chapter Name Triangles Exercise Ex 7.1 Number of Questions Solved 8 Category NCERT Solutions for Class 9 Maths Chapter 7 Triangles Ex 7.1 Ex 7.1 Class 9 Maths Question 1. In a quadrilateral ACBD, AC = AD, and AB bisects ∠ A (see figure). Show that Δ ABC ≅ ΔABD. What can you say about and BD? Solution: Given : In quadrilateral ACBD. AC = AD and ∠BAC = ∠DAB To prove : ΔABC = ΔABD Proof: In ΔABC and ΔABD, AC = AD [given] ∠BAC = ∠DAB [given] and B = AB [common side] Hence, ΔABC = ΔABD [by SAS congrunce rule] Then, BC = BD Thus, BC and BD are equal. Ex 7.1 Class 9 Maths Question 2. ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see figure). Prove that (i) Δ ABD ≅ Δ BAC (ii) BD = AC (iii) Δ ABD = Δ BAC Solution: Given : In quadrilateral ABCD. AD = BC and ∠DAB = ∠CBA To prove : (i) ΔABD ≅ ΔBAC (ii) BD= AC (iii) ∠ABD = ∠BAC Proof : (i) In ΔABD and ΔBAC, AD-BC [given] ΔDAB = ΔCBA [given] and AB = AB [common side] ∴ Δ ABD ≅ Δ BAC [by SAS congruence rule] (ii) From part (i), Δ ABD ≅ ΔBAC Then, BD = AC [by CPCT] (iii) From part (i), Δ ABD ≅ Δ BAC Then, ∠ABD = ∠BAC [by CPCT] Hence proved. Ex 7.1 Class 9 Maths Question 3. AD and BC are equal perpendiculars to a line segment AB (see figure). Show that CD bisects AB. Solution: Given: AD ⊥ AB and BC ⊥ AB To prove: CD bisects AB. Proof: In Δ AOD and Δ BOC. AD = BC [given] ∠OAD ...