#### Topics

##### Similarity

- Property of three parallel lines and their transversals
- Property of an Angle Bisector of a Triangle
- Basic Proportionality Theorem Or Thales Theorem
- Converse of Basic Proportionality Theorem
- Appolonius Theorem
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Similarity
- Properties of Ratios of Areas of Two Triangles
- Similarity of Triangles
- Similar Triangles
- Similarity Triangle Theorem
- Areas of Two Similar Triangles
- Areas of Similar Triangles

##### Pythagoras Theorem

##### Circle

- Theorem of External Division of Chords
- Theorem of Internal Division of Chords
- Converse of Theorem of the Angle Between Tangent and Secant
- Theorem of Angle Between Tangent and Secant
- Converse: If a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic.
- Corollary of Cyclic Quadrilateral Theorem
- Theorem: Opposite angles of a cyclic quadrilateral are supplementary.
- Corollaries of Inscribed Angle Theorem
- Inscribed Angle Theorem
- Intercepted Arc
- Inscribed Angle
- Property of Sum of Measures of Arcs
- Tangent Segment Theorem
- Converse of Tangent Theorem
- Circles Passing Through One, Two, Three Points
- Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers
- Cyclic Properties
- Tangent - Secant Theorem
- Cyclic Quadrilateral
- Angle Subtended by the Arc to the Point on the Circle
- Angle Subtended by the Arc to the Centre
- Introduction to an Arc
- Touching Circles
- Number of Tangents from a Point on a Circle
- Tangent to a Circle
- Tangents and Its Properties
- Theorem - Converse of Tangent at Any Point to the Circle is Perpendicular to the Radius
- Number of Tangents from a Point on a Circle
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles

##### Co-ordinate Geometry

##### Geometric Constructions

- To Construct Tangents to a Circle from a Point Outside the Circle.
- Construction of Triangle If the Base, Angle Opposite to It and Either Median Altitude is Given
- Construction of Tangent Without Using Centre
- Construction of Tangents to a Circle
- Construction of Tangent to the Circle from the Point on the Circle
- Basic Geometric Constructions
- Division of a Line Segment

##### Trigonometry

##### Mensuration

#### notes

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle in fig. A quadrilateral is said to be a cyclic quadrilateral, if there is a circle passing through all its four vertices.

`=>`∠A + ∠C = 180°

also ∠B + ∠D = 180°.

#### theorem

**Theorem:** The sum of either pair of opposite angles of a cyclic quadrilateral is 180º. **Given :** A cyclic quadrilateral ABCD . **To Prove :** ∠BAD + ∠BCD = ∠ABC + ∠ADC = 180º. **Construction :** Draw AC and DB.**Proof :** ∠ACB = ∠ ADB and

∠BAC = ∠BDC [Angles in the same segment]

`therefore` ∠ACB + ∠BAC = ∠ADB +∠BDC = ∠ADC

Adding ∠ABC on both the sides, we get

∠ACB + ∠BAC +∠ABC = ∠ADC +∠ABC

But ∠ACB +∠BAC +∠ABC = 180º. [sum of the angles of a triangle]

`therefore` ∠ ADC + ∠ABC =180º.

`therefore` ∠BAD +∠BCD = 360º - (∠ADC + ∠ABC) = 180º.

Hence proved.**Theorem:** If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.

#### Video Tutorials

#### Shaalaa.com | Circles Theorem: Cyclic quadrilateral

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