#### Topics

##### Number Systems

##### Number Systems

##### Algebra

##### Polynomials

##### Linear Equations in Two Variables

##### Coordinate Geometry

##### Geometry

##### Coordinate Geometry

##### Mensuration

##### Introduction to Euclid’S Geometry

##### Lines and Angles

- Introduction to Lines and Angles
- Basic Terms and Definitions
- Intersecting Lines and Non-intersecting Lines
- Parallel Lines
- Pairs of Angles
- Parallel Lines and a Transversal
- Lines Parallel to the Same Line
- Angle Sum Property of a Triangle

##### Statistics and Probability

##### Triangles

##### 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
- 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

##### 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

##### Areas - Heron’S Formula

##### Surface Areas and Volumes

##### Statistics

##### Algebraic Expressions

##### Algebraic Identities

##### Area

##### Constructions

- Introduction of Constructions
- Basic Constructions
- Some Constructions of Triangles

##### Probability

## 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)

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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