By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia
Introduction to Geometry Definition of Geometry Geometry Greek word means (geo = earth, metry = measure) .. Geometry talks about: Points. Lines angles shapes (plane & solid) By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Cont… Point: to specify exact location, they are zero-dimensional. In other words, points have no length, width, or height. .B Line: is set of points which arranged in straight. A line is one-dimensional, having length, but no width or height. Others of lines are; line segment which has two endpoint rays those have one endpoint Angle: space between the two lines. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angle properties Naming angles the middle letter is the vertex of the angle. AOB or BOA both they are same mention O A B Vertex Arm Angle By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Cont…. In the figure below, name all angles using letters: (a) 450 (b) 1350 (c) 1800 O A B C 450 1350 1800 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Types of angles Type of angle Description Acute angle an angle that less than 90˚. Right angle an angle that 90˚ exactly. Obtuse angle an angle that greater than 90˚ but less 1800 Straight angle an angle that is 180˚ exactly. Reflex angle an angle that is greater than 180˚. Revolution an angle that turns exactly Acute Right Obtuse Straight Reflex Full Rotation By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Plane shapes: Plane shapes: are set of lines collected ,they have two-dimensional; length and width, but no height. They are polygons and circles. They have perimeter and area but they do not have volume since they are flat By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Polygons A polygon is a 2-D shape made when line segments enclose a region. A The end points are called vertices. One of these is called a vertex. B The line segments are called sides. E C D These two dimensions are length and width. A polygon has no height. 2-D stands for two-dimensional. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Naming polygons Polygons are named according to the number of sides they have. Number of sides Name of polygon 3 4 5 6 7 8 9 10 Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Types of Triangles Right triangle: Has only one right angle (90 degrees) This box indicates a right angle or a 90-degree angle. 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Types of Triangles According to sides: Equilateral triangle: A triangle with three congruent (equal) sides and three equal angles These marks indicate equality. 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Types of Triangles Isosceles triangle: A triangle with at least two congruent (equal) sides By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Types of Triangles Scalene triangle: A triangle that has no congruent (equal) sides By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Types of Triangles Name each type of triangle 2 4 Right 1 Scalene 3 Equilateral Isosceles By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

quadrilateral : Quadrilateral is four sided polygon. The familiar Quadrilaterals are: Square Rectangle w Rhombus parallelogram l Trapezium By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Solid shapes Solid shape : collection of some plane shapes which makes three dimensional shape having length, width and height. The most common solids are below: cube Sphere cuboid cylinder cone prism pyramid 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Properties of solid shapes
Solid can have: length, wide and height they are called three dimensions volume and surface area Face: is the flat surface Edge: is where two faces meet. Vertex: is a point where edges meet. Plane of symmetry: is the plane which cuts the solid into two equal arts. Net: is a flat shape that can be fold to make a solid 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

NET Cuboid By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Prism Prism is solid which two parallel faces (base & top) is one of plane shapes (triangle, rectangle etc….). These are some properties of prism: The cross – section is same all along its length. Top and base are same shape (plane shapes). Shape of the end faces give the prisms a name such as (triangular prism) Volume (v) = base area × height. Surface area = total faces of its net. 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Prisms which are familiar include:
Triangular prism: base and top are triangle Circular prism: two end faces are circle, in this case becomes cylinder. Rectangular prism: two end faces are rectangle. This included cuboid. Square prism: base and top are square. This can be cube or cuboid Pentagonal prism: two end faces are pentagon. Hexagonal prism: two end faces are hexagon. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Practice By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Pyramid: By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Pyramids which are familiar include
Triangular Pyramid: base is a triangle Circular prism: base is circle, in this case becomes cone. Rectangular prism: base is a rectangle. Square prism: base is a square. Pentagonal prism: base is a pentagon. Hexagonal prism: base is a hexagon. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

figures Vertex Square based pyramid By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Composite solid shapes
A shape contains more than one solid shape we say composite shape look these shapes: This shape is made of prism in base and pyramid at top we add their volumes A shape below contains cylinder at base and cone at the top, if you ask volume and surface area add the volume and surface area of two shapes By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Composite solid shapes
If remove a part of the shape, we subtract that part from the whole shape like this By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Activity A hemisphere of diameter 10 cm is attached to a cylinder of equal diameter as shown. If the total length of the shape is 20 cm calculate: The surface area of the hemisphere. The length of the hemisphere. The surface area of the whole shape. 20 cm 10 cm By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

MENSURATION Mensuration is mathematical name for calculating perimeters, areas, volumes and other geometric parts of standard geometric shapes such as circles, spheres, prisms, cylinders, cones etc……through the use of mathematical formulas By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

formula of shapes.pdf By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angle properties(angle facts)
Angles between lines: angles at a point = Angles on straight line = Vertically opposite angels are ______ Corresponding angles in parallel lines are_____ Alternate angles in parallel lines are______ interior angles in parallel lines = 3600 1800 equal equal equal 1800 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angle Facts(angle properties)
Angles at a Point a b c d Angles at a point add to 360o Angle : a + b + c + d = 3600 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles at a Point Angles at a point is 360o 75o 85o 80o a Example: Find angle a. 85 75 80 + 240 Angle a = ( ) = = 120o By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles on a straight line add to 180o
Angles on a Line 90 Angles on a straight line add to 180o 180o Oblique line Horizontal line a b Angles a + b = 180o 70o b x 35o Angle x = 180 – 35 = 145o Angle b = 180 – 70 = 110o By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Vertically opposite angles
When two lines intersect, two pairs of vertically opposite angles are formed. a b c d a = c and b = d Vertically opposite angles are equal. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

activity f = 360 – ( ) Find all unknown angles f g 60o 45o 120o 110o 35o a b 22o 116o c d f = = 85o a = 180 – 35 = 145o b = 180 – (22+90) = 68o 3g = 360 – (90+60) = 210 g = 70 Opposite angles are equal So c = 116o d = 180 – 116 = 64o By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Complementary angles When two angles add up to 90° they are called complementary angles. a b a + b = 90° Angle a and angle b are complementary angles. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Supplementary angles When two angles add up to 180° they are called supplementary angles. b a a + b = 180° Angle a and angle b are supplementary angles. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Perpendicular & parallel line
Define: Perpendicular Lines Two lines at right angles (90o) to each other Define: Parallel Straight lines that are always the same distance apart and never meet By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles made with parallel lines
When a straight line crosses two parallel lines eight angles are formed. a b d c e f h g Transversal By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Corresponding angles There are four pairs of corresponding angles, or F-angles. a b d d c e f h h g d = h because Corresponding angles are equal By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Corresponding angles There are four pairs of corresponding angles, or F-angles. a a b b d d c c e e f f h h g g a = e because Corresponding angles are equal By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Corresponding angles There are four pairs of corresponding angles, or F-angles. a b d c c e f h g g c = g because Corresponding angles are equal By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Corresponding angles There are four pairs of corresponding angles, or F-angles. a b b d c e f f h g b = f because Corresponding angles are equal By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Alternate angles There are two pairs of alternate angles, or Z-angles. a b d d c e f f h g d = f because Alternate angles are equal By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Alternate angles There are two pairs of alternate angles, or Z-angles. a b d c c e e f h g c = e because Alternate angles are equal By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Interior angles Interior angles between parallel lines are supplementary. a b b d c e f f h g d + e = 180 because They are interior angles By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Activity Now do these: 65o p 77o 38o 130o y x s r q Corresponding angles p = 65o x = 130o Alternate angles q = 38o y = 130o Corresponding angles r = 77o Opposite angles (with r) or Alternate angles with 77o By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia s = 77o 11/25/2020

Activity 1 . Which one of the following angles is not acute: a) 79˚ b) 89˚ c) 91˚ d) 1˚ 2 . Which one of the following angles is not obtuse? a) 91˚ b) 149˚ c) 179˚ d) 182˚ 3. In figure below, name the angle which: corresponding to AXP, BXY, QYD, and CYQ? Alternate to BXY, and XYD? A B C D P Q x y By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

4.Find the sizes of the lettered angles, and give the reason: a b c 70 o d e f g h i 400 a b 140o h k g k 1050 j c d) By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Sum of the interior angles in a polygon
quadrilateral can be divided into two triangles … … and a pentagon can be divided into three triangles. A hexagon can be divided into four triangles. How many triangles can a hexagon be divided into? By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles in polygons Sum of angles in triangle = sum of angles of any quadrilateral = interior angles of any polygons = Exterior angels of any polygons = Base angles of isosceles angles are______ Exterior angle of a triangle equals the sum of opposite______________ 1800 3600 (n – 2 )1800 3600 equal Interior angles By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

The angles in a triangle add up to 180°.
b c For any triangle, a + b + c = 180° The angles in a triangle add up to 180°. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Calculating angles in a triangle
Calculate the size of the missing angles in each of the following triangles. 64° b 116° 33° a 31° 326° 82° 49° 43° 25° d 88° c 28° 233° 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Angles in an isosceles triangle
In an isosceles triangle, two of the sides are equal. We indicate the equal sides by drawing dashes on them. The two angles at the bottom on the equal sides are called base angles. The two base angles are also equal. If we are told one angle in an isosceles triangle we can work out the other two. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles in an isosceles triangle
For example, Find the size of the other two angles. 88° 46° 46° a a The two unknown angles are equal so call them both a. We can use the fact that the angles in a triangle add up to 180° to write an equation. 88° + a + a = 180° 88° + 2a = 180° 2a = 92° a = 46° 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Practice : Find the value of the lettered angles and give reasons 1. 2. 3. x 62° A B C D Diagram accurately drawn NOT 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Exterior angles in triangle
The angles inside a polygon are called interior angles. When we extend the sides of a polygon outside the shape exterior angles are formed. e d f By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Exterior Angle in a Triangle
Opposite interior angles x = a b a Exterior angle b c X The measure of an exterior angle of a triangle equals the sum of the two remote interior angles. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Calculating angles Calculate the size of the lettered angles in each of the following triangles. 116° b 33° a 64° 82° 31° 34° 43° c 25° d 152° 131° 127° 272° By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Sum of the interior angles in a polygon
The number of triangles that a polygon can be divided into is always two less than the number of sides. We can say that: A polygon with n sides can be divided into (n – 2) triangles. The sum of the interior angles in a triangle is 180°. So, The sum of the interior angles in an n-sided polygon is: S = (n – 2) × 180°. Where s = sum of angles n = number of sides By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Example (1) Find the sum of the interior angles of a regular pentagon, and hence, the size of each interior angle. Solution For a pentagon, n = 5, therefore the sum of the interior angles S = (5-2) 180˚ = 180 × 3˚ = 540˚ For a regular pentagon the interior angle are of equal size. Therefore each angle = 540˚÷5 =108˚. 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Example (2) The sum of the angles of a polygons is 1980˚ how many sides has the polygon? Solution S = (n-2) x180˚ 1980˚÷180˚ = (n-2) x 180˚÷180˚ 11 = n-2 n = 11+2 = 13 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Example (3) The sum of 7 angles of a nonagon is 1000˚, the other two angles are equal so find both sizes Solution S = (n - 2) ×180˚ = (9-2) × 180˚ = 7 × 180˚ = 1260˚ Let the other two angles 2x. 2x = 1260˚-1000˚ 2x = 260˚ x =130˚ By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Activity 1) Find the sum of the interior angles of a hexagon ( number of sides = 6 In hexagon since n = 6 S = (n – 2 ) 1800 S = (6 – 2 ) 1800 S = 4 x 1800 S = 7200 2) If the sum of angles of a polygon is Find the number of sides the polygon has, hence give its name S = (n – 2 ) 1800 8400 = (n – 2 ) 1800 Divide both sides by 1800 n = 9 hence polygon is a nonagon By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Exterior angle of a polygon
An exterior angle of a polygon is formed by extending one side of the polygon. Angle CDY is an exterior angle D E Y B C A F 1 2 Exterior Angle + Interior Angle of a polygon =1800 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

1200 600 1200 600 600 1200 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

1200 1200 1200 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

1200 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020 Polygons

600 600 600 600 600 600 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020 Polygons

1 2 3 4 5 6 600 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020 Polygons

3 4 2 3600 5 1 6 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020 Polygons

900 900 900 900 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020 Polygons

900 900 900 900 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia Polygons

3 2 3600 4 1 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020 Polygons

The sum of the exterior angle of a polygon For any convex polygon the sum of the exterior angles is 360˚. F A B C D E If the polygon is regular and has n-sides, then each exterior angle = 3600 / n No. of sides = 3600/ exterior angle Example : - a) Find the size and exterior angle of a regular nonagon 360˚÷9 = 40˚. b) Calculate the number of sides a regular polygon has, if each exterior angle is 15˚. n = 3600 ÷150 = 24. The polygon has 24 sides. By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

more problems Find the measure of each interior angle of a polygon with 9 sides. Ans : 1400 Find the measure of each exterior angle of a regular decagon. Ans : 360 How many sides are there in a regular polygon if each interior angle measures 1650? Ans : 24 sides Is it possible to have a regular polygon with an exterior angle equal to 400 ? Ans : Yes By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia Polygons 11/25/2020

Angles in Circles (circle geometry) Parts of a circle Minor arc circumference Minor segment Minor Sector Radius diameter Major segment Major Sector Major arc A line drawn at right angles to the radius at the circumference is called the Tangent By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles in Circles Key words: Subtend: an angle subtended by an arc is one whose two rays pass through the end points of the arc end points of the arc arc two rays angle subtended by the arc Supplementary: two angles are supplementary if they add up to 180o Cyclic Quadrilateral: a quadrilateral whose 4 vertices lie on the circumference of a circle 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Fact- 1 : Tangent & Radius
A tangent is perpendicular to the radius of a circle By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Fact -2 :Two tangents from a point outside circle
Tangents are equal PA = PB PO bisects angle APB 90° <APO = <BPO g g <PAO = <PBO = 90° 90° AO = BO (Radii) The two Triangles APO and BPO are Congruent By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles in Circles Theorem 1: The angle subtended in a semicircle is a right angle By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles in Circles Now do these: a 70o b 45o c 130o x 2x a = 180-(90+70) a = 20o b = 180-(90+45) b = 45o = 50o 3x = x = 30o c = 180-(90+50) c = 40o By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles in Circles Theorem 2: The angle subtended by an arc at the centre of a circle is twice that at the circumference This can also appear like or 2a a a 2a arc a 2a arc By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles in Circles Theorem 3: Angles subtended by the same arc (or chord) are equal same arc same angle same arc same angle By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Angles in Circles h = 96 × 2 h = 192o Now do these: d 80o f 78o 72o e g 67o 96o h d = 80 ÷ 2 d = 40o e = 72 × 2 e = 144o g = g = 147.5 f = 78 ÷ 2 f = 39o 67 ÷ 2 = 33.5o By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Remember Theorem 1 The angle in a semi-circle is a right angle. o Diameter o a b c 70o d 30o e angle a = angle b = angle c = angle d = angle e = 90o 90o 20o 90o 60o 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Remember Theorem 2 The angle subtended by an arc at the centre of a circle is twice that at the circumference. xo o 2xo 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Find the unknown angles giving reasons for your answers.
Remember o A B 42o xo 1 Find the unknown angles giving reasons for your answers. po 2 62o yo qo angle x = angle y = (180 – 2 x 42) = 96o (Isos triangle/angle sum triangle). 48o (Angle at the centre) angle p = angle q = 124o (Angle at the centre) (180 – 124)/2 = 280 (Isos triangle/angle sum triangle).

remember Theorem 3 Angles subtended by an arc or chord in the same segment are equal. yo xo xo By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

xo yo 30o 40o Find the unknown angles in each case Angle x = angle y = 38o Angle x = 30o By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia Angle y = 40o 11/25/2020

Cyclic Quadrilateral Theorem.
The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180o) w x q p y z r s Angles x + w = 180o Angles p + q = 180o Angles y + z = 180o Angles r + s = 180o

Find the missing angles below given reasons in each case.
x y 70o 135o p r q Find the missing angles below given reasons in each case. angle x = angle y angle p = angle q = angle r 180 – 85 = 95o (cyclic quad) 180 – 135 = 45o (straight line) 180 – 110 = 70o (cyclic quad) 180 – 70 = 110o (cyclic quad) 180 – 45 = 135o (cyclic quad)

The Alternate Segment Theorem.
The angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment. 45o xo yo 60o zo Find the missing angles below giving reasons in each case. angle x = angle y = angle z = xo yo yo xo 45o (Alt Seg) 60o (Alt Seg) 75o angle sum triangle By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Chord Bisector Theorem. Theorem 6 A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord.. O O S T 3 cm 8 cm Find length OS OS = 5 cm (pythag theorem: 3,4,5) 11/25/2020 By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

Chord Bisector Theorem.
A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord.. O Find angle x O S T 22o xo U Angle SOT = 22o (symmetry/congruenncy) Angle x = 180 – 112 = 68o (angle sum triangle) 11/25/2020

Angles in Circles More complex problems G H C d e 65o f 28o 64o a b c B X O D F I A E The angle subtended in a semicircle is a right angle The angle subtended by an arc at the centre of a circle is twice that at the circumference a = 90o d = 130o Cyclic quadrilateral ACDE The angle subtended by an arc at the centre of a circle is twice that at the circumference Angle AED is supplementary to angle ACD b = 180 – 64 = 116o or Cyclic quadrilateral ABDE Angles subtended by the same arc (or chord) are equal Angle ABD is supplementary to angle AED b = 180 – 116 = 64o e = 65o Opposite angles are equal, therefore triangles FGX and HXI are congruent. f = 28o By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Home work PTR is a tangent line to the circle at T. Find angles SUT, SOT, OTS and OST. Angle SUT = Acute Angle SOT Angle OTS = Angle OST = 65o (Alt seg) 130o (angle at centre) 25o (tan rad) 25o (isos triangle) By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

Classify the figures indicates theorems: A B O C A B C D A B C2 C1 C3 A B D C A Y X A Y X O X B O A A T B Q P B M O C By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia 11/25/2020

By Math Teacher: Hamed Noor C/O : Nawawi Gardo, Puntland, Somalia

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