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
Quadrilaterals: A four-sided polygon is a quadrilateral. Any four points in a plane, out of which three are non-collinear are joined in order to formed a four-sided closed figure called ‘quadrilateral’.
- Any four points in a plane, out of which three are non-collinear are joined in order to formed a four-sided closed figure called ‘quadrilateral’.
- A quadrilateral has four sides, four angles and four vertices. Quadrilateral could be regular or irregular.
A four-sided polygon is a quadrilateral.
- Quadrilateral ABCD has four sides `bar"AB", bar"BC", bar"CD", and bar"DA"`.
- It has four angles ∠A, ∠B, ∠C, and ∠D.
- A, B, C and D are the four vertices and
- BD and AC are the two diagonals of the quadrilateral ABCD.
Reading and Writing of a Quadrilateral:
A quadrilateral can be named by starting at any vertex and going serially either clockwise or anti-clockwise around the figure.
When writing the name of a quadrilateral a sign like this ‘□’ is put in place of the word ‘quadrilateral’.
|Quadrilateral MNOP||□ MNOP||Quadrilateral PONM||□ PONM|
|Quadrilateral NOPM||□ NOPM||Quadrilateral ONMP||□ ONMP|
|Quadrilateral OPMN||□ OPMN||Quadrilateral NMPO||□ NMPO|
|Quadrilateral PMNO||□ PMNO||Quadrilateral MPON||□ MPON|
1. Adjacent Sides of a Quadrilateral:
- Adjacent sides of the quadrilateral have a common vertex.
- The sides MN and MP of □ MNOP have a common vertex M. Sides MN and MP are adjacent sides.
- Every quadrilateral has four pairs of adjacent sides.
Pairs of adjacent sides:
Side MN and Side MP
Side MN and Side NO
Side NO and Side OP
Side OP and Side MP.
2. Opposite Sides of a Quadrilateral:
- Opposite sides of the quadrilateral do not have a common vertex.
- In □ MNOP the sides MP and NO have no common vertex.
- Side MP and side NO are opposite sides of the quadrilateral.
Pairs of opposite sides:
sides MP and NO
sides MN and PO
3. Adjacent Angles of a Quadrilateral:
- The angles of a quadrilateral which have one common arm are called adjacent angles of the quadrilateral.
- These angles are neighbouring or adjacent angles.
Name the adjacent angles of the quadrilateral MNOP.
∠MNO and ∠PMN
∠MPO and ∠NOP
∠PON and ∠MNO
∠ONM and ∠PMN
4. Opposite Angles of a Quadrilateral:
- The angles of a quadrilateral which do not have a common arm are called opposite angles of a quadrilateral.
- They lie opposite to each other.
Pair of Opposite angle:
- Angle opposite to ∠PMN is ∠NOP.
- Angle opposite to ∠MNO is ∠OPM.
5. Diagonals of a Quadrilateral:
- The line segments which join the vertices of the opposite angles of a quadrilateral are the diagonals of the quadrilateral.
- The segments MO and NP are the diagonals of the quadrilateral ABCD.
6. Interior & Exterior of A Quadrilateral:
- Being a polygon, a quadrilateral has an exterior and an interior.
- P, Q, and R are in the interior of the quadrilateral, M and L are in the exterior, and A, B, C, D, and E on the quadrilateral.
Shaalaa.com | Quadrilateral
The angles A, B, C and D of a quadrilateral are in the ratio 2:3: 2 : 3. Show this quadrilateral is a parallelogram.
Three angles of a quadrilateral are equal. If the fourth angle is 69°; find the measure of equal angles.
Two angles of a quadrilateral are 89° and 113°. If the other two angles are equal; find the equal angles.
Two angles of a quadrilateral are 68° and 76°. If the other two angles are in the ratio 5 : 7; find the measure of each of them.
Angles of a quadrilateral are (4x)°, 5(x+2)°, (7x – 20)° and 6(x+3)°. Find :
(i) the value of x.
(ii) each angle of the quadrilateral.
In quadrilateral ABCD, side AB is parallel to side DC. If ∠A : ∠D = 1 : 2 and ∠C : ∠B = 4 : 5
(i) Calculate each angle of the quadrilateral.
(ii) Assign a special name to quadrilateral ABCD