Geometry

GIKPKC7 94107

Geometry I

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Introduction 11/4/98

Terms: Interval Parallel () Triangle () Quadrilateral Bisects Median Altitude Hypotenuse Congruent () Similar () Complementary Supplementary RTP

Line Equidistant from each other A supplementary three sided figure Four sided figure, whose sum of the interior angles is 360 Equal either side of the bisected figure Middle Horizontal The side opposite to the right angle (longest side) in a right-angled triangle Two figures that have the same shape and size Two figures of the same shape but different size Their sum is 90 Their sum is 180 Respects to proof

Notation:      

One letter (E.g. B) is a point Two letters (E.g. BA or AB) is a line or interval Parallel lines are written (E.g. ABCD) Angles are written (E.g.  BAC, BAC or BAC) A triangle is written (E.g. ABC) A quadrilateral is called (E.g. ABCD).

Types of Angles:        

Acute angle 0 < x < 90 Right angle 90 Obtuse angle 90 < x < 180 Straight angle 180 Reflex angle 180 < x < 360 Revolution 360 Vertically opposite angles are equal Exterior angle of a triangle is the sum of the two opposite interior angles

Luke Cole

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GIKPKC7 94107

Geometry I

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Types of Triangles:      

Scalene triangle Right-angled triangle Isosceles triangle Equilateral triangle Acute-angled triangle Obtuse-angled triangle

No equal sides Contains a right angle and hypotenuse Two equal sides & two equal opposite angles Three equal sides All the angles are acute Contains one obtuse angle

When a Transversal Line Cuts Through Two Parallel Lines:

Transversal

Alternate Angles  

Alternate angles form a Z shape Alternate angles are equal

Corresponding Angles  

Corresponding angles form an F shape Corresponding angles are equal

Co-interior Angles  

Co-interior angles form a C shape The sum of the co-interior angles is 180

Luke Cole

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GIKPKC7 94107

Geometry I

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Congruent Triangles: A

X

C

B

Z

Y

Here, ABC  XYZ

Tests    

SSS: SAS: AAS: RHS:

All three pairs of corresponding Sides are equal. Two pairs of corresponding Sides and their included Angles are equal. Two pairs of Angles and one pair of corresponding Sides are equal. Both have a Right angle, their Hypotenuses are equal and one other pair of corresponding Sides are equal.

Similar Triangles: A X C

B

Z

Y

Here, ABC  XYZ

Tests   

All three pairs of corresponding angles are equal. All three pairs of corresponding sides are in proportion (in the same ratio). Two pairs of side are in proportion and their included angles are equal.

Ratio of Intercepts: 

When two or more transversals cut a series of parallel lines, the ratios of their intercepts are equal. A D B C

E F

AB:BC = DE:EF AB DE  BC EF Luke Cole

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GIKPKC7 94107

Geometry I

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Polygons:

 A polygon is a plane figure with straight sides.  A regular polygon has all sides and all interior angles equal. Equation: S = (2.n – 4)  90 S = Sum of the interior angles n = number of sides  The sum of the exterior angles of any polygon is 360. Exterior Interior

Pythagoras’ Theorem:

 Only right-angled triangles. Equation: c2 = a2 + b2 c = Hypotenuse a = 1st side b = 2nd side Proof: Let, AD = x DB = y  x+y=c ADC  ABC (AAA) AC AD  AB AC b x  c b  b2 = c.x BDC  ABC (AAA) DB BC  BC AB  a2 = c.y  a2 + b2 = c.y + c.x = c(y + x) = c(c) 2 2  a + b = c2

A c b

C

Luke Cole

D

a

B

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GIKPKC7 94107

Geometry I

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Types of Quadrilaterals 11/4/98

Parallelogram: 

A parallelogram is a quadrilateral with opposite sides parallel

Properties    

Opposite sides are equal Opposite angles are equal Diagonals bisect each other Each diagonal bisects the parallelogram into two congruent triangles

Tests    

Both pairs of opposite sides are equal Both pairs of opposite angles are equal One pair of sides is both equal and parallel The diagonals bisect each other

Rectangle: 

A rectangle is a parallelogram with one angle a right angle

Properties  

The same as for a parallelogram Diagonals are equal

Test 

Diagonals are equal

Luke Cole

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GIKPKC7 94107

Geometry I

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Rhombus: 

A rhombus is a parallelogram with a pair of adjacent sides equal

Properties   

The same as for a parallelogram Diagonals bisect at right angles Diagonal bisect the angles of the rhombus

Tests  

All sides are equal Diagonals bisect each other at right angles

Square: 

A square is a rectangle with a pair of adjacent sides equal

Properties   

The same as for a rectangle Diagonals are perpendicular Diagonals make angles of 45 with the sides

Trapezium: 

A trapezium is a quadrilateral with one pair of sides parallel

Kite: 

A kite is a quadrilateral with two pairs of adjacent side’s equal

Luke Cole

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GIKPKC7 94107

Geometry I

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Areas 13/4/98

Rectangle: Equation:

A = l.b l b

Square: Equation:

A = x2 x

Triangle: Equation:

A = ½.b.h h b

Parallelogram: Equation:

A = b.h h b

Rhombus: Equation:

A = ½.x.y x

y

Luke Cole

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GIKPKC7 94107

Geometry I

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Trapezium: Equation:

A = ½.h(a + b) a h b

Circle: Equation:

A = .r2 r

Luke Cole

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