A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 +...
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Transcript of A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 +...
![Page 1: A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 + ∠ 4 = ∠ C...... (ii) We know that the sum of the angles.](https://reader036.fdocuments.us/reader036/viewer/2022062718/56649ea95503460f94bae13e/html5/thumbnails/1.jpg)
QUADRILATERALS
CH - 8
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A Quadrilateral is a figure which is formed by joining four points in an order is called a quadrilateral.
Each figure is formed by Joining 4 points.
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![Page 3: A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 + ∠ 4 = ∠ C...... (ii) We know that the sum of the angles.](https://reader036.fdocuments.us/reader036/viewer/2022062718/56649ea95503460f94bae13e/html5/thumbnails/3.jpg)
Sides – 4
A
B C
D
Angles sum – 360
Diagonals - 2
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![Page 4: A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 + ∠ 4 = ∠ C...... (ii) We know that the sum of the angles.](https://reader036.fdocuments.us/reader036/viewer/2022062718/56649ea95503460f94bae13e/html5/thumbnails/4.jpg)
Angle Sum Property of a Quadrilateral
Let ABCD be a quadrilateral. Join AC.
Clearly, 1 + 2 = A ...... (i) ∠ ∠ ∠
And, 3 + 4 = C ...... (ii) ∠ ∠ ∠
We know that the sum of the angles of a triangle is 180°.
Angle Sum Property of a Quadrilateral
Therefore, from ∆ABC, we have
∠2 + 4 + B = 180° (Angle sum property of triangle)∠ ∠
From ∆ACD, we have
∠1 + 3 + D = 180° (Angle sum property of triangle) ∠ ∠
Adding the angles on either side, we get;
∠2 + 4 + B + 1 + 3 + D = 360° ∠ ∠ ∠ ∠ ∠
⇒ ( 1 + 2) + B + ( 3 + 4) + D = 360° ∠ ∠ ∠ ∠ ∠ ∠
⇒ ∠A + B + C + D = 360° [using (i) and (ii)]. ∠ ∠ ∠
Hence, the sum of all the four angles of a quadrilateral is 360°.
It states that sum of angles of a quadrilaterals is 360⁰PROOF :-
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Major Types of Quadrilaterals
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PARALLELOGRAM
• Opposite sides are parallel by definition.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• The diagonals bisect each other.
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![Page 7: A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 + ∠ 4 = ∠ C...... (ii) We know that the sum of the angles.](https://reader036.fdocuments.us/reader036/viewer/2022062718/56649ea95503460f94bae13e/html5/thumbnails/7.jpg)
TRAPEZIUM
• One pair of opposite side are parallel.
• Opposite non-parallel sides are equal in length.
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SQUAREA D
B C
•The diagonals of a square bisect each other and meet at 90°•The diagonals of a square bisect its angles.•The diagonals of a square are perpendicular.
•Opposite sides of a square are both parallel and equal in length.•All four angles of a square are equal.•All four sides of a square are equal.
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RHOMBUS
•All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite angles are congruent, and consecutive angles are supplementary).•All sides are congruent by definition.•The diagonals bisect the angles.
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![Page 10: A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 + ∠ 4 = ∠ C...... (ii) We know that the sum of the angles.](https://reader036.fdocuments.us/reader036/viewer/2022062718/56649ea95503460f94bae13e/html5/thumbnails/10.jpg)
Rectangle
•All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite sides are congruent, and diagonals bisect each other).•All angles are right angles by definition.•The diagonals are congruent
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![Page 11: A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 + ∠ 4 = ∠ C...... (ii) We know that the sum of the angles.](https://reader036.fdocuments.us/reader036/viewer/2022062718/56649ea95503460f94bae13e/html5/thumbnails/11.jpg)
KITE
• The diagonals of a kite meet at a right angle.
• Kites have exactly one pair of opposite angles that are congruent.
• The Opposite sides are parallel.
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Note-A square, rectangle and rhombus are all parallelograms.A square is a rectangle and also a rhombus.A parallelogram is a trapezium.A kite is not a parallelogram.A trapezium is not a parallelogram(as only one pair of opposite sides is parallel in a trapezium and we require both pairs to be parallel in a parallelogram).A rectangle or a rhombus is not a square.
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----------Theorems----------• A diagonal of a parallelogram divides it
into two congruent triangles.• In a parallelogram, opposite sides are
equal.• If each pair of opposite sides of a
quadrilateral is equal, then it is a parallelogram.
• In a parallelogram, opposite angles are equal.
• If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
• The diagonals of a parallelogram bisect each other.
• If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
• A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.
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![Page 14: A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 + ∠ 4 = ∠ C...... (ii) We know that the sum of the angles.](https://reader036.fdocuments.us/reader036/viewer/2022062718/56649ea95503460f94bae13e/html5/thumbnails/14.jpg)
THE ENDAyushword.wordpress.com