Basic Concepts in Geometry
- Concept of Parallel Lines
- Checking Parallel Lines
- Pairs of Lines - Transversal of Parallel Lines
- Properties of Angles Formed by Two Parallel Lines and a Transversal
- Axiom: If a Transversal Intersects Two Parallel Lines, Then Each Pair of Interior Angles on the Same Side of the Transversal is Supplementary.
- Axiom: If a Transversal Intersects Two Parallel Lines, Then Each Pair of Corresponding Angles is Equal.
- Axiom: If a Transversal Intersects Two Parallel Lines, Then Each Pair of Alternate Interior Angles Are Equal.
- Use of properties of parallel lines
- Test for Parallel Lines
- Interior Angles Test
- Alternate Angles Test
- Corresponding Angles Test
- Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Theorem of remote interior angles of a triangle
- Exterior Angle of a Triangle and Its Property
- Congruence of Triangles
- Isosceles Triangles : If Two Sides of a Triangle Are Equal, the Angles Opposite to Them Are Also Equal.
- Isosceles Triangle : If Two Angles of a Triangle Are Equal, the Sides Opposite to Them Are Also Equal.
- Property of 30°- 60°- 90° Triangle Theorem
- Property of 45°- 45°- 90° Triangle Theorem
- Median of a 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.
- 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.
- 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: The diagonals of a square are perpendicular bisectors of each other.
- Property: The diagonals of a rhombus are perpendicular bisectors of one another.
- Properties of Isosceles Trapezium
- The Mid-point Theorem
- Converse of Mid-point Theroem
- 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.
- Construction of the Circumcircle of a Triangle
- Circumference of a Circle
Surface area and volume
In the figure, point D and E are on side BC of Δ ABC, such that BD = CE and AD = AE.
Show that Δ ABD ≅ Δ ACE.
In the given figure, seg PT is the bisector of ∠ QPR. A line through R intersects ray QP at point S. Prove that PS = PR
In the given figure, ∠ RST = 56° , seg PT ⊥ ray ST, seg PR ⊥ ray SR and seg PR ≅ seg PT Find the measure of ∠ RSP. State the reason for your answer.
Prove that, if the bisector of ∠ BAC of Δ ABC is perpendicular to side BC, then Δ ABC is an isosceles triangle.