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Class 1: Angles Class 2: Parallel lines and angles Class 3:
Quadrilaterals and types of triangles.Class 4: Congruent triangles.
Class 5: Theorems 1- 4Class 6: Theorems 5 & 6 Class 8: Theorem
8Class 9: Theorem 9 Class 10: Theorem 10 MenuSelect the class
required then clickmouse key to view class.Class 7: Theorem 7 and
the three deductions.(Two classes is advised)
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AnglesAn angle is formed when two lines meet. The size of the
angle measures the amount of space between the lines. In the
diagram the lines ba and bc are called the arms of the angle, and
the point b at which they meet is called the vertex of the angle.
An angle is denoted by the symbol .An angle can be named in one of
the three ways:
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1. Three lettersUsing three letters, with the centre at the
vertex. The angle is now referred to as : abc or cba.
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2. A numberPutting a number at the vertex of the angle. The
angle is now referred to as 1.
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3. A capital letterPutting a capital letter at the vertex of the
angle.The angle is now referred to as B.
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Right angleA quarter of a revolution is called a right
angle.Therefore a right angle is 90. Straight angleA half a
revolution or two right angles makes a straight angle.A straight
angle is 180.Measuring angles
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Acute, Obtuse and reflex AnglesAny angle that is less than 90 is
called an acute angle.An angle that is greater than 90 but less
than 180 is called an obtuse angle.An angle greater than 180 is
called a reflex angle.
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Angles on a straight line Angles on a straight line add up to
180. A + B = 180 . Angles at a point Angles at a point add up to
360.A+ B + C + D + E = 360
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Pairs of lines: Consider the lines L and K :Intersecting
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Parallel linesL is parallel to KWritten: LKParallel lines never
meet and are usually indicated by arrows.Parallel lines always
remain the same distance apart.
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Perpendicular L is perpendicular to K Written: L K
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Now work on practical examples in your maths book.
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Parallel lines and Angles1.Vertically opposite angles When two
straight lines cross, four angles are formed. The two angles that
are opposite each other are called vertically opposite angles. Thus
a and b are vertically opposite angles. So also are the angles c
and d.From the above diagram: A+ B = 180 .. Straight angle B + C =
180 ... Straight angle A + C = B + C Now subtract c from both sides
A = B
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2. Corresponding AnglesThe diagram below shows a line L and four
other parallel lines intersecting it.The line L intersects each of
these lines.LAll the highlighted angles are in corresponding
positions.These angles are known as corresponding angles.If you
measure these angles you will find that they are all equal.
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Remember: When a third line intersects two parallel lines the
corresponding angles are equal.
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3. Alternate angles The diagram shows a line L intersecting two
parallel lines A and B. The highlighted angles are between the
parallel lines and on alternate sides of the line L. These shaded
angles are called alternate angles and are equal in size. Remember
the Z shape.
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Now work on practical examples from your maths books.
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QuadrilateralsA quadrilateral is a four sided figure.The four
angles of a quadrilateral sum to 360.a + b + c + d = 360(This is
because a quadrilateral can be divided up into two triangles.)Note:
Opposite angles in a cyclic quadrilateral sum to 180. a + c = 180b
+ d = 180
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The following are different types of Quadrilaterals
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Parallelogram1. Opposite sides are parallel2. Opposite sides are
equal3. Opposite angles are equal4. Diagonals bisect each other
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Rhombus1. Opposite sides are parallel2. All sides are equal3.
Opposite angles are equal4. Diagonals bisect each other5. Diagonal
intersects at right angles6. Diagonals bisect opposite angles
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Rectangle1. Opposite sides are parallel2. Opposite sides are
equal3. All angles are right angles4. Diagonals are equal and
bisect each other
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Square1. Opposite sides are parallel2. All sides are equal3. All
angles are right angles4. Diagonals are equal and bisect each
other5. Diagonals intersect at right angles6. Diagonals bisect each
angle
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Types of TrianglesEquilateral Triangle3 equal sides3 equal
anglesIsosceles Triangle2 sides equalBase angles are equal a =
b(base angles are the angles opposite equal sides)Scalene triangle3
unequal sides 3 unequal angles
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Now work on practical examples from your maths books.
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Congruent trianglesCongruent means identical. Two triangles are
said to be congruent if they have equal lengths of sides, equal
angles, and equal areas. If placed on top of each other they would
cover each other exactly.The symbol for congruence is . For two
triangles to be congruent (identical), the three sides and three
angles of one triangle must be equal to the three sides and three
angles of the other triangle. The following are the tests for
congruency.
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Case 1=Three sides of the other triangleThree sides of one
triangleSSS Three sides
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Case 2Two sides and the included angle of one triangleTwo sides
and the included angle of one triangle=SAS(side, angle, side)
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Case 3One side and two angles of one triangleCorresponding side
and two angles of one triangle=ASA(angle, side, angle)
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Case 4A right angle, the hypotenuse and the other side of one
triangleA right angle, the hypotenuse and the other side of one
triangle=RHS(Right angle, hypotenuse, side)
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Now do practical examples on congruent triangles in your maths
book.
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Theorem: Vertically opposite angles are equal in measure. Given:
To prove : Construction:Proof: Straight angle Straight angle 1=2
Label angle 31=2Intersecting lines L and K, with vertically
opposite angles 1 and 2. 1+3=180 2+3=180Q.E.D. 1+3=3+2.....Subtract
3 from both sides 3
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Theorem: The measure of the three angles of a triangle sum to
180.Given: To Prove: 1+2+3=180Construction:Proof: 1=4 and
2=5Alternate angles1+2+3=4+5+3But 4+5+3=180Straight angle
1+2+3=180The triangle abc with 1,2 and 3.45Q.E.D.Draw a line
through a, Parallel to bc. Label angles 4 and 5.
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Theorem: An exterior angle of a triangle equals the sum of the
two interior opposite angles in measure.Given:A triangle with
interior opposite angles 1 and 2 and the exterior angle 3. To
prove:1+ 2= 3Construction:Label angle 4Proof:1+ 2+ 4=1803+
4=180Three angles in a triangle 1+ 2+ 4= 3+ 4Straight angle 1+ 2=
34Q.E.D.
Interior opposire
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Theorem: If to sides of a triangle are equal in measure, then
the angles opposite these sides are equal in measure.Given:The
triangle abc, with ab = ac and base angles 1 and 2.To prove:1 =
2Construction:Draw ad, the bisector of bac. Label angles 3 and 4.
Proof: ab = ac given3 = 4construction ad = ad commonSAS 1 =
2Corresponding anglesdQ.E.D.
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Now work on practical examples from your maths books.
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Theorem: Opposite sides and opposite angles of a parallelgram
are respectively equal in measure.Given:Parallelogram abcdTo prove:
Construction:Join a to c. Label angles 1,2,3 and 4.Proof:1= 2 and
3= 4Alternate angles ac = accommonASA ab = dcand ad = bc
Corresponding sidesAnd abc = adcCorresponding anglesSimilarly, bad
= bcd ab = dc , ad = bcabc = adc, bad = bcdQ.E.D.
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Theorem:A diagonal bisects the area of a
parallelogram.Given:Parallelogram abcd with diagonal [ac].To
prove:Area of abc = area of adc.Proof: ab = dcOpposite sides ad =
bc Opposite sides ac = acCommon SSSQ.E.D.
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Now work on practical examples from your maths books.
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Theorem: The measure of the angle at the centre of the circle is
twice the measure of the angle at the circumference, standing on
the same arc.Given: Circle, centre o, containing points a, b and
c.To prove: boc = 2 bacConstruction: Join a to o and continue to d.
Label angles 1,2,3,4 and 5.Proof: d1= 2 + 3Exterior angleBut 2 = 3
1 = 2 2Similarly, 5 = 2 4 1+ 5 = 2 2 + 2 4 1 + 5 = 2(2 + 4) i.e.
boc = 2 bac Q.E.D.
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Deduction 1: All angles at the circumference on the same arc are
equal in measure.To prove: bac = bdcProof:3 = 2 1Angle at the
centre is twice the angle on the circumference (both on the arc
bc)3 = 2 2 Angle at the centre is twice the angle on the
circumference (both on arc bc) 2 1 = 2 2 1 = 2 i.e. bac =
bdcQ.E.D.
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Deduction 2: An angle subtended by a diameter at the
circumference is a right angle.To prove: bac = 90Proof:2 = 2 1Angle
at the centre is twice the angle on the circumference (both on the
arc bc) straight line. But 2 = 180 2 1 = 180 1 = 90 i.e. bac =
90Q.E.D.
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Deduction 3: The sum of the opposite angles of a cyclic
quadrilateral is 180.To prove: bad + bcd = 180 3 = 2 1Proof:Angle
at the centre is twice the angle on the circumference. (both on
minor arc bd)4 = 2 2 Angle at the centre is twice the angle on the
circumference. (Both on the major arc bd) 3 + 4 = 2 1 + 2 2But 3 +
4 = 360Angles at a point 2 1 + 2 2 = 360 1 + 2 = 180i.e. bad + bcd
= 180Q.E.D.
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Now work on practical examples from your maths books.
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Theorem: A line through the centre of a circle perpendicular to
a chord bisects the chord. Given: Circle, centre c, a line L
containing c, chord [ab], such that L ab and L ab = d.To prove: ad
= bdConstruction:Label right angles 1 and 2.Proof:1 = 2 = 90 Given
ca = cb Both radii cd = cd commonR H S Corresponding
sidesQ.E.D.
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Now work on practical examples from your maths books.
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Theorem: If two triangles are equiangular, the lengths of the
corresponding sides are in proportion.Given : Two triangles with
equal angles.To prove: Construction: On ab mark off ax equal in
length to de. On ac mark off ay equal to df and label the angles 4
and 5.Proof:1 = 4[xy] is parallel to [bc]As xy is parallel to
bc.Q.E.D.
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Now work on practical examples from your maths books.
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Theorem: In a right-angled triangle, the square of the length of
the side opposite to the right angle is equal to the sum of the
squares of the other two sides. Q.E.D.To prove that angle 1 is
90Proof:3+ 4+ 5 = 180 Angles in a triangle But 5 = 90 => 3+ 4 =
90 => 3+ 2 = 90 Since 2 = 4 Now 1+ 2+ 3 = 180 Straight line=>
1 = 180 - ( 3+ 2 )=> 1 = 180 - ( 90 )Since 3+ 2 already proved
to be 90=> 1 = 90
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Now work on practical examples from your maths books.
Interior opposire