#### Topics

##### Basic Concepts in Geometry

- Introduction to Basic Concepts in Geometry
- Concept of Points
- Concept of Line
- Concept of Plane
- Co-ordinates of Points and Distance
- Betweenness
- Concept of Line Segment
- Concept of Ray
- Conditional Statements and Converse
- Proofs

##### Parallel Line

- Parallel Lines
- Checking Parallel Lines
- Pairs of Lines - Transversal of Parallel Lines
- Properties of Parallel Lines
- Interior Angle Theorem
- Corresponding Angle Theorem
- Alternate Angles Theorems
- Use of properties of parallel lines
- Test for Parallel Lines
- Interior Angles Test
- Alternate Angles Test
- Corresponding Angles Test
- Corollary of Parallel Lines

##### Triangles

- Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Remote Interior Angles of a Triangle Theorem
- Exterior Angle of a Triangle and Its Property
- Congruence of Triangles
- Isosceles Triangles Theorem
- Converse of Isosceles Triangle Theorem
- Corollary of a Triangle
- Property of 30°- 60°- 90° Triangle Theorem
- Property of 45°- 45°- 90° Triangle Theorem
- Median of a Triangle
- Property of Median Drawn on the Hypotenuse of Right Triangle
- Perpendicular Bisector Theorem
- Angle Bisector Theorem
- Properties of inequalities of sides and angles of a triangle
- Similar Triangles
- Similarity of Triangles

##### Constructions of Triangles

- Perpendicular Bisector Theorem
- Construction of Triangles
- To Construct a Triangle When Its Base, an Angle Adjacent to the Base, and the Sum of the Lengths of Remaining Sides is Given.
- To Construct a Triangle When Its Base, Angle Adjacent to the Base and Difference Between the Remaining Sides is Given.
- To Construct a Triangle, If Its Perimeter, Base and the Angles Which Include the Base Are Given.

##### Quadrilaterals

- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- Properties of a Parallelogram
- Properties of Rhombus
- Properties of a Square
- Properties of Rectangle
- Properties of Trapezium
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Property: The adjacent angles in a parallelogram are supplementary.
- Tests for Parallelogram
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram
- Theorem: If One Pair of Opposite Sides of a Quadrilateral Are Equal and Parallel, It is a Parallelogram.
- Property: The Diagonals of a Rectangle Are of Equal Length.
- Property: Diagonals of a Square Are Congruent.
- Property: The diagonals of a square are perpendicular bisectors of each other.
- Property: Diagonals of a Square Bisect Its Opposite Angles.
- Property: The diagonals of a rhombus are perpendicular bisectors of one another.
- Property: Diagonals of a Rhombus Bisect Its Opposite Angles.
- Properties of Isosceles Trapezium
- Theorem of Midpoints of Two Sides of a Triangle
- Converse of Mid-point Theorem

##### Circle

- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Properties of Chord
- Theorem: a Perpendicular Drawn from the Centre of a Circle on Its Chord Bisects the Chord.
- Theorem : The Segment Joining the Centre of a Circle and the Midpoint of Its Chord is Perpendicular to the Chord.
- Relation Between Congruent Chords of a Circle and Their Distances from the Centre
- Properties of Congruent Chords
- Theorem: Equal chords of a circle are equidistant from the centre.
- Theorem : The Chords of a Circle Which Are Equidistant from the Centre Are Equal.
- Incircle of a Triangle
- Construction of the Incircle of a Triangle.
- Circumcentre of a Triangle
- Construction of the Circumcircle of a Triangle

##### Co-ordinate Geometry

- Coordinate Geometry
- The Co-ordinates of a Point in a Plane
- Co-ordinates of Points on the Axes
- Plotting a Point in the Plane If Its Coordinates Are Given.
- Equations of Lines Parallel to the X-axis and Y-axis
- Graphs of Linear Equations
- The Graph of a Linear Equation in the General Form

##### Trigonometry

- Trigonometry
- Terms Related to Right Angled Triangle
- Trigonometric Ratios and Its Reciprocal
- Relation Among Trigonometric Ratios
- Trigonometric Ratios of 30° and 60° Angles
- Trigonometric Table
- Important Equation in Trigonometry

##### Surface area and volume

**Theorem:**If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

## Theorem

**If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.**

This is also referred to as the **corresponding angles axiom**.

**Given:** Two Parallel lines PQ and RS.

Let AB be the transversal intersecting PQ at M and RS and N.

**To Prove: **Each pair of corresponding angles are equal.

i.e., ∠ AMP ≅ ∠ MNR, ∠ PMN ≅ ∠ RNB,

and ∠ AMQ ≅ ∠ MNS, ∠ QMN ≅ ∠ SNB.

**Proof: **

First, we will prove ∠ AMP ≅ ∠ MNR.

For lines PQ and RS with transversal AB,

∠QMN = ∠MNR ......(Alternate Interior angles)(1)

For lines PQ and AB,

∠AMP = ∠QMN .......(Vertically opposite angles)(2)

From (1) and (2),

∠AMP = ∠MNR

Similarly, we can prove that

∠ PMN ≅ ∠ RNB,

∠ AMQ ≅ ∠ MNS,

∠ QMN ≅ ∠ SNB.

Hence, Each pair of corresponding angles are equal.

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