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Menu. Class 1: Angles. Class 2: Parallel lines and angles . Class 3: Quadrilaterals and types of triangles. Class 4: Congruent triangles. Class 5: Theorems 1- 4. Class 6: Theorems 5 & 6. Class 7: Theorem 7 and the three deductions.(Two classes is advised). Class 8: Theorem 8.
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Menu Class 1: Angles Class 2: Parallel lines and angles Class 3: Quadrilaterals and types of triangles. Class 4: Congruent triangles. Class 5: Theorems 1- 4 Class 6: Theorems 5 & 6 Class 7: Theorem 7 and the three deductions.(Two classes is advised) Class 8: Theorem 8 Class 9: Theorem 9 Class 10: Theorem 10 Select the class required then click mouse key to view class.
. a . Amount of space Angle b c Angles An 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:
. a . b c 1. Three letters Using three letters, with the centre at the vertex. The angle is now referred to as : abc or cba.
. a . 1 b c 2. A number Putting a number at the vertex of the angle. The angle is now referred to as 1.
. a . B b c 3. A capital letter Putting a capital letter at the vertex of the angle. The angle is now referred to as B.
We use the symbol to denote a right angle. Measuring angles Right angle A quarter of a revolution is called a right angle. Therefore a right angle is 90. Straight angle A half a revolution or two right angles makes a straight angle. A straight angle is 180.
Acute, Obtuse and reflex Angles Any 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.
A B B C A D E 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
L . p K L intersects K at p written : L K = {p} Pairs of lines: Intersecting Consider the lines L and K :
L K Parallel lines L is parallel to K Written: LK Parallel lines never meet and are usually indicated by arrows. Parallel lines always remain the same distance apart.
L K The symbol is placed where two lines meet to show that they are perpendicular Perpendicular L is perpendicular to K Written: L K
C A B D Parallel lines and Angles 1.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 Angles The diagram below shows a line L and four other parallel lines intersecting it. L The line L intersects each of these lines. All 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.
In the given diagram the line L intersects two parallel lines A and B. The highlighted angles are equal because they are corresponding angles. The angles marked with are also corresponding angles L . . A . B Remember: When a third line intersects two parallel lines the corresponding angles are equal.
L A B 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.
b c a d Quadrilaterals A 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 = 180 b + d = 180
. .. .. . Parallelogram 1. Opposite sides are parallel 2. Opposite sides are equal 3. Opposite angles are equal 4. Diagonals bisect each other
.. . . .. .. . .. . Rhombus 1. Opposite sides are parallel 2. All sides are equal 3. Opposite angles are equal 4. Diagonals bisect each other 6. Diagonals bisect opposite angles 5. Diagonal intersects at right angles
Rectangle 2. Opposite sides are equal 1. Opposite sides are parallel 4. Diagonals are equal and bisect each other 3. All angles are right angles
. . . . . . . . Square 2. All sides are equal 3. All angles are right angles 1. Opposite sides are parallel 4. Diagonals are equal and bisect each other 5. Diagonals intersect at right angles 6. Diagonals bisect each angle
. . . a b Types of Triangles Isosceles Triangle Equilateral Triangle 3 equal sides 2 sides equal 3 equal angles Base angles are equal a = b Scalene triangle (base angles are the angles opposite equal sides) 3 unequal sides 3 unequal angles
x a abc xyz z c b y Congruent triangles Congruent 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 one triangle Three sides of the other triangle SSS Three sides
Case 2 = Two sides and the included angle of one triangle Two sides and the included angle of one triangle SAS (side, angle, side)
Case 3 Corresponding side and two angles of one triangle = One side and two angles of one triangle ASA (angle, side, angle)
Case 4 = A right angle, the hypotenuse and the other side of one triangle A 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.
L 2 1 K Theorem: Vertically opposite angles are equal in measure. Given: Intersecting lines L and K, with vertically opposite angles 1 and 2. 3 To prove : 1=2 Construction: Label angle 3 Proof: 1+3=180 Straight angle 2+3=180 Straight angle 1+3=3+2 .....Subtract 3 from both sides 1=2 Q.E.D.
a 3 1 2 c b Theorem: The measure of the three angles of a triangle sum to 180. Given: The triangle abc with 1,2 and 3. To Prove: 1+2+3=180 4 5 Construction: Draw a line through a, Parallel to bc. Label angles 4 and 5. Proof: 1=4 and 2=5 Alternate angles 1+2+3=4+5+3 But 4+5+3=180 Straight angle 1+2+3=180 Q.E.D.
a 1 2 3 b c Theorem: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. 4 A triangle with interior opposite angles 1 and 2 and the exterior angle 3. Given: To prove: 1+ 2= 3 Construction: Label angle 4 Three angles in a triangle Proof: 1+ 2+ 4=180 3+ 4=180 Straight angle 1+ 2+ 4= 3+ 4 Q.E.D. 1+ 2= 3
a 3 4 2 1 b c Consider abd and acd: abd acd Theorem: If to sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure. d The triangle abc, with ab = ac and base angles 1 and 2. Given: To prove: 1 = 2 Construction: Draw ad, the bisector of bac. Label angles 3 and 4. Proof: given ab = ac construction 3 = 4 common ad = ad SAS Corresponding angles 1 = 2 Q.E.D.
a d 4 1 2 b 3 c Consider abc and adc : abc adc Theorem: Opposite sides and opposite angles of a parallelgram are respectively equal in measure. Parallelogram abcd Given: ab = dc , ad = bc To prove: abc = adc, bad = bcd Construction: Join a to c. Label angles 1,2,3 and 4. Proof: Alternate angles 1= 2 and 3= 4 common ac = ac ASA Corresponding sides ab = dcand ad = bc Corresponding angles And abc = adc Similarly, bad = bcd Q.E.D.
a d b c Consider abc and adc: abc adc area abc = area adc Theorem:A diagonal bisects the area of a parallelogram. Parallelogram abcd with diagonal [ac]. Given: Area of abc = area of adc. To prove: Proof: Opposite sides ab = dc Opposite sides ad = bc Common ac = ac SSS Q.E.D.
a 2 4 . o 1 5 3 c b Consider aob: Base angles in an isosceles 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. d Circle, centre o, containing points a, b and c. Given: boc = 2 bac To prove: Construction: Join a to o and continue to d. Label angles 1,2,3,4 and 5. Proof: Exterior angle 1= 2 + 3 But 2 = 3 1 = 22 Similarly, 5 = 2 4 1+ 5 = 2 2 + 2 4 1 + 5 = 2(2 + 4) Q.E.D. i.e. boc = 2 bac
d a 2 1 . o 3 c b Deduction 1: All angles at the circumference on the same arc are equal in measure. To prove: bac = bdc Proof: 3 = 2 1 Angle 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 = bdc Q.E.D.
a 1 . o b c 2 Deduction 2: An angle subtended by a diameter at the circumference is a right angle. bac = 90 To prove: Angle at the centre is twice the angle on the circumference (both on the arc bc) straight line. 2 = 2 1 Proof: But 2 = 180 2 1 = 180 1 = 90 i.e. bac = 90 Q.E.D.
a 1 . 4 o d 3 b 2 c Deduction 3: The sum of the opposite angles of a cyclic quadrilateral is 180. To prove: bad + bcd = 180 Angle at the centre is twice the angle on the circumference. (both on minor arc bd) Proof: 3 = 2 1 Angle at the centre is twice the angle on the circumference. (Both on the major arc bd) 4 = 2 2 3 + 4 = 2 1 + 2 2 Angles at a point But 3 + 4 = 360 2 1 + 2 2 = 360 i.e. bad + bcd = 180 1 + 2 = 180 Q.E.D.
a . ∟ d 1 L c ∟ 2 b Consider cda and cdb: cda cdb ad = bd 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 = bd Construction: Label right angles 1 and 2. Proof: Given 1 = 2 = 90 Both radii ca = cb common cd = cd R H S Corresponding sides Q.E.D.
|ab| |ac| |ab| |ab| |ac| |ac| = = = |bc| |bc| |de| |df| |de| |ax| |ay| |df| = = |ef| d |ef| a 2 2 5 4 1 3 x y f e Similarly 1 3 c b Theorem: If two triangles are equiangular, the lengths of the corresponding sides are in proportion. Two triangles with equal angles. Given : To prove: 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. Construction: Proof: 1 = 4 [xy] is parallel to [bc] As xy is parallel to bc. Q.E.D.
b a a c 2 b c 1 3 c b c a 5 4 a b 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. To prove that angle 1 is 90º Proof: 3+ 4+ 5 = 180º ……Angles in a triangle But 5 = 90º => 3+ 4 = 90º => 3+ 2 = 90º ……Since 2 = 4 Now 1+ 2+ 3 = 180º ……Straight line => 1 = 180º - ( 3+ 2 ) => 1 = 180º - ( 90º ) ……Since 3+ 2 already proved to be 90º => 1 = 90º Q.E.D.
