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Ex 11.1, 2 - Construct angle 45 degree - Class 9 Constructions

Ex 11.1, 2 - Chapter 11 Class 9 Constructions - Part 2
Ex 11.1, 2 - Chapter 11 Class 9 Constructions - Part 3
Ex 11.1, 2 - Chapter 11 Class 9 Constructions - Part 4

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Ex 11.1, 2 - Chapter 11 Class 9 Constructions - Part 5

Ex 11.1, 2 - Chapter 11 Class 9 Constructions - Part 6
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Ex 11.1, 2 Construct an angle of 45° at the initial point of a given ray and justify the construction . Steps of construction Draw a ray OA. Taking O as center and any radius, draw an arc cutting OA at B. 3. Now, taking B as center and with the same radius as before, draw an arc intersecting the previously drawn arc at point C. 4. With C as center and the same radius, draw an arc cutting the arc at D 5. With C and D as centers and radius more than 1/2 CD, draw two arcs intersecting at P. 6. Join OP. Thus, ∠ AOP = 90° Now we draw bisector of ∠ AOP 7. Let OP intersect the original arc at point Q 8. Now, taking B and Q as centers, and radius greater than 1/2 BQ, draw two arcs intersecting at R. 9. Join OR. Thus, ∠ AOR = 45° Justification We need to prove ∠ AOR = 45° Join OC & OB Thus, OB = BC = OC ∴ Δ OCB is an equilateral triangle ∴ ∠ BOC = 60° Join OD, OC and CD Thus, OD = OC = DC ∴ Δ DOC is an equilateral triangle ∴ ∠ DOC = 60° Join PD and PC Now, In Δ ODP and Δ OCP OD = OC DP = CP OP = OP ∴ Δ ODP ≅ Δ OCP ∴ ∠ DOP = ∠ COP So, we can say that ∠ DOP = ∠ COP = 1/2 ∠ DOC ∠ DOP = ∠ COP = 1/2 × 60° = 30° (Radius of same arcs) (Arc of same radii) (Common) (SSS Congruency) (CPCT) (We proved earlier that ∠ DOC= 60° ) Now, ∠ AOP = ∠ BOC + ∠ COP ∠ AOP = 60° + 30° = 90° Now, Join QR and BR In Δ OQR and Δ OBR OQ = OB QR = BR OR = OR ∴ Δ OQR ≅ Δ OBR ∴ ∠ QOR = ∠ BOR (Radius of same arcs) (Arc of same radii) (Common) (SSS Congruency) ∠ QOR = ∠ BOR = 1/2 ∠ AOP ∠ DOP = ∠ COP = 1/2 × 90° = 45° Thus, ∠ AOR = 45° Hence justified

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.