Question 26 - End-of-Chapter Exercises - Chapter 5 Class 9 - I’m Up and Down, and Round and Round (Ganita Manja

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Question 26 How would you use the following figure to justify the statement that the sum of the opposite angles of a cyclic quadrilateral is 180^∘? Here, Each small triangle (eg: ∆ OBC) is an isosceles triangle And, angles opposite to equal sides are equal So, ∠ OCB = ∠ OBC = q Similarly, we can find all angles And, at the end we can use Angle sum property of Quadrilateral to prove opposite angles are supplementary Let’s do this step-by-step Step 1 of 6 The Setup & Radii We are given a cyclic quadrilateral with the center of its circumcircle at . Lines , and OD are drawn from the center to the vertices. Because they all connect the center to the edge of the same circle, they are all radii. Therefore, . This is marked by the double ticks on each line. Previous Next StepStep 2 of 6 2. Focus on Triangle OAB Let's focus on just one triangle first: . In this triangle, side OA equals side OB (since both are radii). This makes an isosceles triangle. A fundamental property of isosceles triangles is that the angles opposite the equal sides are also equal. Therefore, . Let's label both of these angles as . Previous Next StepStep 3 of 6 3. Triangle OBC Now let's move to the adjacent triangle: . Applying the exact same logic, side OB equals side OC (radii). This makes another isosceles triangle. Therefore, its base angles must be equal: . Let's label these as q. Previous Next StepStep 4 of 6 4. Generalizing the Remaining Triangles We can generalize this property for the remaining two triangles: In , since , the base angles are equal. Let's label them . In , since , the base angles are also equal. Let's label them . Previous Next StepStep 5 of 6 5. The Sum of All Angles Now let's look at the full interior angles of the quadrilateral : The sum of all interior angles in ANY quadrilateral is always . Previous Next StepStep 5 of 6 5. The Sum of All Angles Now let's look at the full interior angles of the quadrilateral : The sum of all interior angles in ANY quadrilateral is always . Previous Next Step Step 5 of 6 5. The Sum of All Angles Now let's look at the full interior angles of the quadrilateral : The sum of all interior angles in ANY quadrilateral is always . Previous Next StepStep 6 of 6 6. Proving the Opposite Angles We now know that . Let's add a pair of opposite angles, for example, and : Rearranging this gives us: . Since we just proved that ( ) , it means that . Similarly, . Therefore, the sum of the opposite angles of a cyclic quadrilateral is always !