Topics

Number Systems
Number Systems
- Introduction of Real Number
- Concept of Irrational Numbers
- Real Numbers and Their Decimal Expansions
- Representing Real Numbers on the Number Line
- Operations on Real Numbers
- Laws of Exponents for Real Numbers
Algebra
Polynomials
- Concept of Polynomials
- Polynomials in One Variable
- Zeroes of a Polynomial
- Remainder Theorem
- Factorisation of Polynomials
- Factorising the Quadratic Polynomial (Trinomial) of the type ax2 + bx + c, a ≠ 0.
- Algebraic Identities
Linear Equations in Two Variables
- Linear Equations in Two Variables
- Linear Equation in One Variable
- Solution of a Linear Equation
- Graph of a Linear Equation in Two Variables
- Equations of Lines Parallel to the X-axis and Y-axis
Algebraic Expressions
- Algebraic Expressions
Algebraic Identities
- Algebraic Identities
Coordinate Geometry
- Coordinate Geometry
- Cartesian System
- Plotting a Point in the Plane If Its Coordinates Are Given.
Geometry
Introduction to Euclid’S Geometry
- Concept for Euclid’S Geometry
- Euclid’S Definitions, Axioms and Postulates
- Equivalent Versions of Euclid’S Fifth Postulate
Lines and Angles
- Introduction to Lines and Angles
- Basic Terms and Definitions
- Intersecting Lines and Non-intersecting Lines
- Concept of Parallel Lines
- Pairs of Angles
- Parallel Lines and a Transversal
- Lines Parallel to the Same Line
- Angle Sum Property of a Triangle
Triangles
- Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Congruence of Triangles
- Criteria for Congruence of Triangles
- Properties of a Triangle
- Some More Criteria for Congruence of Triangles
- Inequalities in a Triangle
Quadrilaterals
- 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
- The Mid-point Theorem
- 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
Area
- Concept of Area
- Corollary: A rectangle and a parallelogram on the same base and between the same parallels are equal in area.
- Theorem: Parallelograms on the Same Base and Between the Same Parallels.
- Corollary: Triangles on the same base and between the same parallels are equal in area.
Circles
- 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
Constructions
- Introduction of Constructions
- Basic Constructions
- Some Constructions of Triangles
Mensuration
Areas - Heron’S Formula
- Area of a Triangle
- Area of a Triangle by Heron's Formula
- Application of Heron’s Formula in Finding Areas of Quadrilaterals
Surface Areas and Volumes
- Surface Area of a Cuboid
- Surface Area of a Cube
- Surface Area of Cylinder
- Surface Area of a Right Circular Cone
- Surface Area of a Sphere
- Volume of a Cuboid
- Volume of a Cylinder
- Volume of a Right Circular Cone
- Volume of a Sphere
Statistics and Probability
Statistics
- Concepts of Statistics
- Collecting Data
- Presentation of Data
- Graphical Representation of Data
- Measures of Central Tendency
Probability
- Probability - an Experimental Approach

notes

The opposite angles of a parallelogram are of equal measure.

Given: ABCD is a parallelogram.

To Prove: m∠ A = m∠C

Construction: Draw diagonal AC and diagonal BD.

To Prove:

AC and BD are the diagonals of the parallelogram,

∠ BAC = ∠ ACD and ∠ DAC = ∠ ACB......(pair of alternate angles) and `bar(AC)` is common side,

In ΔABC and ∆ADC,

∠ BAC = ∠ ACD ......(pair of alternate interior angles)

∠ DAC = ∠ ACB ......(pair of alternate interior angles)

Side AC = Side AC ......(common side)

∆ABC ≅ ∆CDA ......(ASA congruency criterion)

This shows that ∠B and ∠D have same measure. In the same way you can get, m∠A = m ∠C.

Alternatively, ∠ BAC = ∠ ACD and ∠ DAC = ∠ ACB

We have, m∠ A = ∠ BAC + ∠ DAC and ∠ ACB + ∠ ACD = m∠C.

m∠A = m ∠C.

Hence Proved.

Example

In the given Fig, BEST is a parallelogram. Find the values x, y, and z.

S is opposite to B.

So,
x = 100° ........(opposite angles property)

y = 100° ..........(a measure of angle corresponding to ∠ x)

z = 80° ..........(since ∠y, ∠z is a linear pair)

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Related concepts

Properties of a Parallelogram - Property: The Opposite Sides of a Parallelogram Are of Equal Length.

Properties of a Parallelogram - Property: The adjacent angles in a parallelogram are supplementary.

Properties of a Parallelogram - Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.

Properties of a Parallelogram - Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.

Properties of a Parallelogram - Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)

Properties of a Parallelogram - Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram

Properties of a Parallelogram - Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.

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Properties of a Parallelogram - Property: The Opposite Angles of a Parallelogram Are of Equal Measure.

Topics

Number Systems

Number Systems

Algebra

Polynomials

Linear Equations in Two Variables

Algebraic Expressions

Algebraic Identities

Coordinate Geometry

Geometry

Introduction to Euclid’S Geometry

Lines and Angles

Triangles

Quadrilaterals

Area

Circles

Constructions

Mensuration

Areas - Heron’S Formula

Surface Areas and Volumes

Statistics and Probability

Statistics

Probability

notes

The opposite angles of a parallelogram are of equal measure.

Example

Shaalaa.com | To Prove that the Opposite Angles of a Parallelogram are equal in measure

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Properties of a Parallelogram - Property: The Opposite Angles of a Parallelogram Are of Equal Measure.

Topics

Number Systems

Number Systems

Algebra

Polynomials

Linear Equations in Two Variables

Algebraic Expressions

Algebraic Identities

Coordinate Geometry

Geometry

Introduction to Euclid’S Geometry

Lines and Angles

Triangles

Quadrilaterals

Area

Circles

Constructions

Mensuration

Areas - Heron’S Formula

Surface Areas and Volumes

Statistics and Probability

Statistics

Probability

notes

The opposite angles of a parallelogram are of equal measure.

Example

Shaalaa.com | To Prove that the Opposite Angles of a Parallelogram are equal in measure

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Series: Property: The Opposite Angles of a Parallelogram Are of Equal Measure.

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