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GEOMETRY TYPES OF ANGLES 1. ACUTE ANGLE An angle which is less than 90º 2. RIGHT ANGLE An angle is 90º 3. OBTUSE-ANGLED An angle between 90º and 180º 4. REFLEX ANGLE An angle greater than 180º 5. COMPLEMENTARY ANGLES a These are angles whose sum is 90º b e.g. a + b = 90º 6. SUPPLEMENTARY ANGLES These are angles whose sum is 180º e.g. a + b = 180º 1 b a PROPERTIES OF ANGLES & STRAIGHT LINES Consider a straight line: 180º The total angle on this line is 180º. If another straight line is drawn and meets the other line at some point “p”, then the two angles adjacent to each other “a” and “b” will sum to 180º. They are also known as SUPPLEMENTARY ANGLES. b a p INTERSECTION OF TWO STRAIGHT LINES When two straight lines intersect each other, then the angles opposite each other at the intersection are equal in value. ∠a = ∠c and ∠b = ∠d c b d a PARALLEL LINES CUT BY A TRANSVERSAL The arrows on the horizontal lines show they are in parallel, therefore the corresponding angles are equal: ∠a = ∠w , ∠b = ∠x , ∠c = ∠y , ∠d = ∠z a d Now as ∠a = ∠c and ∠b = ∠d then ∠c = ∠w and ∠d = ∠x (i.e. alternate angles) Also as ∠a = ∠w and ∠b = ∠d , then ∠w + ∠d = 180º and ∠c + ∠x = 180º These are known as supplementary angles 2 w z y x b c PROOF OF PARALLEL LINES If two straight lines are cut by a transversal, then the two lines are parallel if any of the following conditions apply: 1. Two corresponding angles are equal. ∠a = ∠w 2. Two alternate angles are equal. ∠d = ∠x 3. Two interior angles are supplementary. ∠c + ∠x = 180º Ex 1: Calculate the value of angle ά if line BF bisects ∠ABC . D θ = 80º γ A B β φ = 38º ά F C X 1. 2. 3. 4. 5. 6. E Z Y Lines DX, BZ & EY are parallel to each other as they are all at 90º to line XY. ∠DAB = ∠ABZ or θ = γ = 80º (i.e. alternate angles) β = φ = 38º (i.e. alternate angles) ∠ABC = γ + β = 80º+38º=118º As line FB bisects angle ∠ABC , then angle ∠FBC = 59º β + α = 59º, hence α = 59º − β = 59º − 38º = 21º 3 Ex 2: In the diagram below, prove that ∠x = ∠a + ∠b . b (alternate angle) a x 180º- (a + b) b a (alternate angle) Sum of all the angles around the cross = 360º ∴ 360º = 2 x + 2 {180º − ( a + b )} ∴ 2 x = 360º − 2 {180º − ( a + b )} ∴ x = 180º − {180º − ( a + b )} = 180º − 180º + ( a + b ) = a + b ∴∠x = ∠a + ∠b when the two lines are in parallel. a a x x b b ∠x = ∠a + ∠b ∠x = ∠a + ∠b 4 TRIANGLES TYPES OF TRIANGLES 1. ACUTE-ANGLED All angles are less than 90º 2. RIGHT-ANGLED One angle is 90º The side opposite this angle is known as the hypotenuse 3. OBTUSE-ANGLED Has one angle greater than 90º 4. EQUILATERAL 60º All its sides and angles are equal, hence each angle = 60º 60º 60º 5. ISOSCELES a b Two sides and two angles are equal. N.B. The equal angles are opposite the equal sides N.B. The sum of all the angles in any triangles = 180º 5 B A CONGRUENT TRIANGLES Consider the following two triangles: A D B C E F Side AB = Side DE Side BC = Side EF Side AC = Side DF Also angles: Λ Λ Λ A=D Λ Λ B=E Λ C=F As these triangles are equal in every respect, then they are said to be congruent. Also if 2 triangles are congruent, then their areas are equal. i.e. ΔABC = ΔDEF Although two triangles may be congruent, it is not necessary to prove that all the sides and angles are equal. The following may be used to prove congruency: 1. One side and 2 angles are equal (similar location) A B D E C 6 F 2. Two sides and an angle between them are equal. D A B E C F 3. Three sides of one triangle equal three sides of the other. D A B E C F 4. The hypotenuse and one side of a right-angled triangle are equal. A B D E C 7 F N.B. If 3 angles on one triangle are equal to the angles on the other triangle, this does not prove congruency as a minimum of one side is required. By moving the hypotenuse “DF” to the left, it can be seen that the angles are still the same but the triangles are not congruent. D A B E C F Also two sides and a non-included angle do not prove congruency. In triangles OBA and OCA, angle θ and side OA are common. Also sides OB and OC are equal, however the triangles are not congruent. See in diagram below: C B θ O 8 A PYTHAGORAS THEOREM ( AC ) 2 A 2 = ( AB ) + ( BC ) 2 ∴ AC = 2 ( AB ) + ( BC ) 2 B First of all, this theorem is only applicable to right-angled triangles. It can be seen that if you take the length of each side adjacent to the right angle, square them and then add them together, the resulting sum will equal the longest side (hypotenuse) squared. Example A ladder is placed against a vertical wall that makes a right angle with the horizontal ground. The base of the ladder is 1m away from the wall and the top is at a height of 4m. Calculate the length of the ladder. T 4m 1m TB = B 2 2 ( 4 ) + (1) TB = 16 + 1 = 17 = 4.12m 9 C SIMILAR TRIANGLES Triangles that have the same angles as each other are considered to be similar even though the lengths of the sides are different. Consider the example below. Both triangles are right-angled. The value of θ in the smaller triangle is 90o - 60o = 30o. In the larger triangle, the value of φ is 90o - 30o = 60o, so the triangles are similar, hence the sides of “a” and “b” in the larger triangle can be calculated. 30o θo b 69.282cm 10cm φo 60o 5cm a Firstly, the unknown side of the smaller triangle needs to be calculated using Pythagoras Theorem: 102 − 52 = 75 = 8.66cm The ratio of the larger triangle to the smaller one is: 69.282 =8 8.66 As the ratio of the larger triangle to the smaller one, then side “a” = 8 x 5 = 40cms and side “b” = 8 x 10 = 80cm. N.B. You can also use Pythagoras Theorem to calculate the length of a side once you have the lengths of the other two sides. e.g. b = 402 + 69.2822 = 6400 = 80cm 10