If ∠DAC = 32º and ∠AOB = 70º, then ∠DBC is equal to 38º The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. If ∠DAC = 32º and ∠AOB = 70º, then ∠DBC is equal to, a. NCERT Exemplar Class 9 Maths Exercise 8.1 Problem 12 The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. ☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8 If ∠SPR = 50º and ∠POQ = 80º, then ∠SQR is equal to a. ✦ Try This: The diagonals PR and QS of a parallelogram PQRS intersect each other at the point O. If ∠DAC = 32º and ∠AOB = 70º, then ∠DBC is equal to Corresponding sides are equal, so AB = CD and BC = DA.The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. Thus by ASA, triangles ABC and CDA are congruent. Also line AC is a transversal of parallel lines BC and DA, so angle ACB is congruent to angle CAD. The line AC is a transversal of parallel lines AB and CD, so angle CAB is congruent to angle ACD. To prove the angles congruent, we use transversals. The two triangles have a common side AC = CA. We will prove that triangle ABC is congruent to triangle CDA by ASA. By definition, line AB is parallel to line CD and line BC is parallel to line DA. Proposition: If ABCD is a parallelogram, its opposite sides are equal. (Opposite sides of a parallelogram are equal.) This says ABCD is a rhombus, by definition. From this is follows that the hypotenuses are all congruent: AB = AD = CB = CD. Thus the triangles AMB, AMD, CMB, and CMD are congruent by SAS. If we also assume that AC is perpendicular to BC, then each of the angles AMB, AMD, CMB, and CMD are right angles. We know from this that MA = MC and MB = MD. Let M be the intersection of the diagonals. Proof: From Problem 1, we know that the diagonals of a parallelogram ABCD bisect each other. (b) Prove that a parallelogram with perpendicular diagonals is a rhombus. Proof: In the homework, it was proved that if a quadrilateral ABCD has opposite sides equal, then it is a parallelogram.
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