# Class 10 Math Angles in a Segment of a Circle 1. Prove that in a given cyclic rilateral sum of opposite angles is two right angles and coiversely.

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##### 1. Prove that in a given cyclic rilateral sum of opposite angles is two right angles and coiversely.

(ix) In the figure O is the centre of the circle then the angle x is:(a) 15^{\circ} (b) 30^{\circ} (c) 45^{\circ} (d) 60^{\circ}

(iii) In the adjacent figure if m \angle 3=75^{\circ} then find m \angle 1 and m \angle 2 .(a) 37 \frac{1}{2} 37 \frac{1}{2}^{\circ} (b) 37 \frac{1}{2}^{\circ} 75^{\circ} (c) 75^{\circ} 37 \frac{1}{2}^{\circ} (d) 75^{\circ} 75^{\circ}

(i) A circle passes through the vertices of a right angled \triangle A B C with m \overline{A C}=3 \mathrm{~cm} and m \overline{B C}=4 \mathrm{~cm} m \angle C=90^{\circ} . Radius of the circle is:(a) 1.5 \mathrm{~cm} (b) 2.0 \mathrm{~cm} (c) 2.5 \mathrm{~cm} (d) 3.5 \mathrm{~cm}

3. A O B and C O D are two intersecting chords of a circle. Show that \triangle^{s} A O D and B O C are equiangular.

4. \overline{A D} and \overline{B C} are two parallel chords of a circle. Prove that arc A B \cong \operatorname{arc} C D and arc A C \cong \operatorname{arc} B D .

(ii) In the adjacent circular figure central and inscribed angles stand on the same arc A B . Then(a) m \angle 1=m \angle 2 (b) m \angle 1=2 m \angle 2 (c) m \angle 2=3 m \angle 1 (d) m \angle 2=2 m \angle 1

Example 1: Two equal circles intersect in A and B . Through B a straight line is drawn to meet the circumferences at P and Q respectively. Prove that m \overline{A P}=m \overline{A Q} .

(viii) In the figure O is the centre of the circle then angle x is:(a) 15^{\circ} (b) 30^{\circ} (c) 45^{\circ} (d) 60^{\circ}

(iv) Given that O is the centre of the circle. The angle marked x will be:(a) 12 \frac{1}{2}^{\circ} (b) 25^{\circ} (c) 50^{\circ} (d) 75^{\circ}

(x) In the figure O is the centre of the circle then the angle x is:(a) 50^{\circ} (b) 75^{\circ} (c) 100^{\circ} (d) 125^{\circ}

(vii) In the figure O is the centre of the circle then the angle x is:(a) 55^{\circ} (b) 110^{\circ} (c) 220^{\circ} (d) 125^{\circ}

Example 2: A B C D is a rilateral circumscribed about a circle.Show that m \overline{A B}+m \overline{C D}=m \overline{B C}+m \overline{D A}

2 . Show that parallelogram inscribed in a circle will be a rectangle.

(v) Given that O is the centre of the circle the angle marked y will be:(a) 12 \frac{1}{2}^{\circ} (b) 25^{\circ} (c) 50^{\circ} (d) 75^{\circ}