Class 9 Maths Chapter 9 Circles – Exercise 9.5 NCERT Solutions

Introduction

Exercise 9.5 focuses on properties of cyclic quadrilaterals. Students learn how to prove and apply theorems related to quadrilaterals inscribed in circles, especially the relationship between opposite angles. This exercise strengthens reasoning and proof skills in geometry.

Key Theorem

1. Opposite Angles of a Cyclic Quadrilateral: If a quadrilateral is inscribed in a circle, then the sum of its opposite angles is ${180}^{\circ }$.

$$\mathrm{\angle }A+\mathrm{\angle }C={180}^{\circ },\mathrm{\angle }B+\mathrm{\angle }D={180}^{\circ }$$

2. Converse: If the sum of opposite angles of a quadrilateral is ${180}^{\circ }$, then the quadrilateral is cyclic.

Common Mistakes

• Forgetting to use the property of arcs subtending angles at the circle.

• Confusing cyclic quadrilaterals with general quadrilaterals.

• Not applying the converse theorem correctly.

• Arithmetic errors in angle calculations.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. In a cyclic quadrilateral ABCD, prove that $\mathrm{\angle }A+\mathrm{\angle }C={180}^{\circ }$.

$$\text{Arc }ABC\text{ subtends }\mathrm{\angle }ADC,\text{Arc }CDA\text{ subtends }\mathrm{\angle }ABC$$

$$\mathrm{\angle }ADC+\mathrm{\angle }ABC={180}^{\circ }$$

Q2. In a cyclic quadrilateral PQRS, prove that $\mathrm{\angle }P+\mathrm{\angle }R={180}^{\circ }$. By theorem of cyclic quadrilaterals:

$$\mathrm{\angle }P+\mathrm{\angle }R={180}^{\circ }$$

Q3. In a cyclic quadrilateral, prove that opposite angles are supplementary. Direct application of theorem:

$$\mathrm{\angle }A+\mathrm{\angle }C={180}^{\circ },\mathrm{\angle }B+\mathrm{\angle }D={180}^{\circ }$$

Q4. If one angle of a cyclic quadrilateral is ${90}^{\circ }$, prove that opposite angle is also ${90}^{\circ }$.

$$\mathrm{\angle }A={90}^{\circ }\Rightarrow \mathrm{\angle }C={180}^{\circ }-{90}^{\circ }={90}^{\circ }$$

Q5. Prove that exterior angle of cyclic quadrilateral is equal to interior opposite angle.

$$\mathrm{\angle }E={180}^{\circ }-\mathrm{\angle }D=\mathrm{\angle }B$$

Q6. If opposite angles of a quadrilateral are supplementary, prove that it is cyclic. Converse theorem: Quadrilateral with opposite angles supplementary is cyclic.

Q7. In quadrilateral ABCD, $\mathrm{\angle }A+\mathrm{\angle }C={180}^{\circ }$. Prove that ABCD is cyclic. By converse theorem, quadrilateral is cyclic.

Q8. In quadrilateral PQRS, $\mathrm{\angle }P+\mathrm{\angle }R={180}^{\circ }$. Prove that PQRS is cyclic. By converse theorem, PQRS is cyclic.

Q9. In quadrilateral XYZW, $\mathrm{\angle }X+\mathrm{\angle }Z={180}^{\circ }$. Prove that XYZW is cyclic. By converse theorem, XYZW is cyclic.

Q10. In quadrilateral LMNO, $\mathrm{\angle }L+\mathrm{\angle }N={180}^{\circ }$. Prove that LMNO is cyclic. By converse theorem, LMNO is cyclic.

FAQs (10)

FAQ1. What is cyclic quadrilateral? Quadrilateral inscribed in a circle.

FAQ2. What is property of cyclic quadrilateral? Opposite angles are supplementary.

FAQ3. What is converse theorem? If opposite angles are supplementary, quadrilateral is cyclic.

FAQ4. What is exterior angle property? Exterior angle = interior opposite angle.

FAQ5. Can all quadrilaterals be cyclic? No, only those with opposite angles supplementary.

FAQ6. What is practical use of cyclic quadrilaterals? Used in circle geometry and constructions.

FAQ7. What is supplementary angle? Two angles whose sum is ${180}^{\circ }$.

FAQ8. What is inscribed quadrilateral? Quadrilateral with all vertices on circle.

FAQ9. Why is Exercise 9.5 important? It builds proof skills in geometry.

FAQ10. What is real‑life application of cyclic quadrilaterals? Designs in architecture and engineering.

Conclusion

Exercise 9.5 covers cyclic quadrilaterals with solved examples and FAQs. Mastering these problems helps students in geometry proofs and circle‑related theorems.

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New Syllabus-Class 9 Maths Chapter 9 Circles – Exercise 9.5 NCERT Solutions