Teaching Theorems

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)

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:

1. Three lettersUsing three letters, with the centre at the vertex. The angle is now referred to as : abc or cba.

2. A numberPutting a number at the vertex of the angle. The angle is now referred to as 1.

3. A capital letterPutting a capital letter at the vertex of the angle.The angle is now referred to as B.

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

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.

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

Pairs of lines: Consider the lines L and K :Intersecting

Parallel linesL is parallel to KWritten: LKParallel lines never meet and are usually indicated by arrows.Parallel lines always remain the same distance apart.

Perpendicular L is perpendicular to K Written: L K

Now work on practical examples in your maths book.

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

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.

Remember: When a third line intersects two parallel lines the corresponding angles are equal.

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.

Now work on practical examples from your maths books.

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

The following are different types of Quadrilaterals

Parallelogram1. Opposite sides are parallel2. Opposite sides are equal3. Opposite angles are equal4. Diagonals bisect each other

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

Rectangle1. Opposite sides are parallel2. Opposite sides are equal3. All angles are right angles4. Diagonals are equal and bisect each other

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

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

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.

Case 1=Three sides of the other triangleThree sides of one triangleSSS Three sides

Case 2Two sides and the included angle of one triangleTwo sides and the included angle of one triangle=SAS(side, angle, side)

Case 3One side and two angles of one triangleCorresponding side and two angles of one triangle=ASA(angle, side, angle)

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)

Now do practical examples on congruent triangles in your maths book.

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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


Interior opposire

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Teaching Theorems