Linear Equations in Two Variables
Introduction to Euclid’S Geometry
Lines and Angles
- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- Angle Sum Property of a Quadrilateral
- Types of Quadrilaterals
- Another Condition for a Quadrilateral to Be a Parallelogram
- Theorem of Midpoints of Two Sides of a Triangle
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Angle Subtended by a Chord at a Point
- Perpendicular from the Centre to a Chord
- Circles Passing Through One, Two, Three Points
- Equal Chords and Their Distances from the Centre
- Angle Subtended by an Arc of a Circle
- Cyclic Quadrilateral
Areas - Heron’S Formula
Surface Areas and Volumes
Statistics and Probability
Construct a triangle in which two sides are equal, say each equal to 3.5 cm and the third side equal to 5 cm in following fig.
A triangle in which two sides are equal is called an isosceles triangle. So, ∆ ABC of above Fig. is an isosceles triangle with AB = AC.
Theorem : Angles opposite to equal sides of an isosceles triangle are equal.
Proof : We are given an isosceles triangle ABC in which AB = AC. We need to prove that ∠ B = ∠ C. Let us draw the bisector of ∠ A and let D be the point of intersection of this bisector of
∠ A and BC in following fig.
In ∆ BAD and ∆ CAD,
AB = AC (Given)
∠ BAD = ∠ CAD (By construction)
AD = AD (Common)
So, ∆ BAD ≅ ∆ CAD (By SAS rule)
So, ∠ ABD = ∠ ACD, since they are corresponding angles of congruent triangles.
So, ∠ B = ∠ C
Theorem : The sides opposite to equal angles of a triangle are equal.
This is converse of above theorem.
- The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
Shaalaa.com | Properties of triangle
Which of the following statements are true (T) and which are false (F):
The bisectors of two equal angles of a triangle are equal
Fill in the blank to make the following statement true.
Difference of any two sides of a triangle is........ than the third side.