Deduction 9 - What can we say about the angles in a rhombus?

Transcript

Deduction 9 What can we say about the angles in a rhombus?Let’s consider a rhombus GAME We need to find relation between ∠ 1, ∠ 2, ∠ 3, ∠ 4 Now, Since all sides of rhombus are equal GE = GA = EM = AM In ∆ GEA Since GE = GA By Isosceles triangle property Angles opposite to equal sides are equal ∴ ∠ 1 = ∠ 3 Also, In ∆ GEA Since GE = GA By Isosceles triangle property Angles opposite to equal sides are equal ∴ ∠ 2 = ∠ 4 In ∆ GEA & ∆ MEA GE = MR GA = MA EA = EA ∴ ∆ GEA ≅ ∆ MEA By CPCT (Corresponding Parts of Congruent Triangles) ∠ 1 = ∠ 2 and ∠ 3 = ∠ 4 From (2), (3) and (4) ∠ 1 = ∠ 2 = ∠ 3 = ∠ 4 Let angle be x ∠ 1 = ∠ 2 = ∠ 3 = ∠ 4 = x Also, by CPCT ∠ G = ∠ M Similarly, ∠ E = ∠ A Thus, in a Rhombus, Opposite angles are equal For lines GA and EM With transversal EA ∠ GAE = ∠ MEQ Since alternate angles are equal ∴ GA ∥ EM For lines GE and AM With transversal EA ∠ GEA = ∠ MAE Since alternate angles are equal ∴ GE ∥ AM Thus, Opposite sides of Rhombus are parallel So, is a Parallelogram Therefore, we can write Every Rhombus is a parallelogram Which also means all properties of Parallelogram are also properties of Rhombus