# Line AB (no Endpoints) Ray with Endpoint A. Line Segments with Endpoints A and B. Angle is formed by TWO Rays with a common Endpoint.

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1 Section 8 1 Lines and Angles Point is a specific location in space.. Line is a straight path (infinite number of points). Line Segment is part of a line between TWO points. Ray is part of the line that includes an ENDPOINT. A B.. A B A B..> Point A Line AB (no Endpoints) Ray with Endpoint A Line Segments with Endpoints A and B Angle is formed by TWO Rays with a common Endpoint. ( ) Rays: and Vertex is the common Endpoint. Vertex = B Angle ABC, Angle CBA, Angle B, ABC, CBA Circle = Chapter 8

2 Right Angle = 90 Box at vertex shows 90 Straight Angle = 180 Acute Angle is > 0 and < 90 (small) Obtuse Angle is > 90 and < 180 (Fat) Congruent Angles has the same measure. Complementary Angles measures add to 90. Supplementary Angles measures add to 180. What is the supplement of 35? =145 What is the complement of 52? =38 Parallel Lines never intersect. 2 Chapter 8

3 Two intersecting Lines for four angles. A and C are vertical angles. m A = m C B and D are vertical angles. m B = m D Adjacent Angles share side (Ray). A is adjacent to B and D Perpendicular Lines intersect at 90. When a Line intersects two Parallel Lines, Eight Angles are formed. 1 = 4 = 5 = 8 and 2 = 3 = 6 = 7 Section 8 2 Triangles and the Pythagorean Theorem Triangle is a three sided figure. The sum of the angles of a triangle = Chapter 8

4 What is measure of angle a? X = X = 180 X = = 47 What are the measures of angles a and b? a = 180 a = b = b = 180 b = = 61 Names of Triangles: Acute all angles acute. Right has a right angle. Obtuse has an obtuse angle. Equilateral all three sides equal. Isosceles two sides equal. Scalene NO sides equal. SQUARE ROOTS Square of a number = that number times itself. N 2 = N x N Reverse is the Square Root. 4 Chapter 8

5 = x = N 0 2 = 0 0=0 8 2 = 64 64=8 1 2 = 1 1=1 9 2 = 81 81=9 2 2 = 4 4 = = = = 9 9 = = = = = = = = = = = = = = = = = = = 15 PYTHAGOREAN THEOREM AND APPLICATIONS Right Triangles have a special property. Side a and side b are called legs and are on either side of the 90 angle. Side c is the Hypotenuse and is opposite the 90 angle. (Leg) 2 + (Leg) 2 = (Hypotenuse) 2 a 2 + b 2 = c 2 5 Chapter 8

6 = c = c 2 25 = c 2 25 = 5 = c? = a or b 26 = c a = 26 2 a = 676 a 2 = 100 = 100 a = 10 The bottom of a 17 foot ladder is placed 8 feet from the building. How far up is the ladder on the building? c = 17, a = 8, b = building b 2 = b 2 = 289 b 2 = 225 = 225 b = 15 Section 8 3 Quadrilaterals, Perimeter, and Area Quadrilaterals: 6 Chapter 8

7 Rectangle = Box with four right angles. Opposite sides are equal, adjacent sides are not. Square = Rectangle with all side equal. Parallelogram = four sided figure with two sets of parallel sides. Rhombus = parallelogram with all sides equal. Trapezoid = four sided with ONE set of parallel sides. Perimeter: Square = S + S + S + S = 4S 7 Chapter 8

8 Rectangle = L + W + L + W = 2L + 2W What is the perimeter of a square with a side of 4.3 inches? P = 4(4.3 inches) = 17.2 inches Find the perimeter of a rectangle with a length of 3 yards and a width of 2 feet? P = 2(3 yards)+2(2 feet) P = 6 yards + 4 feet P = 18 feet + 4 feet = 22 feet Area: Rectangle = L x W Square = S x S = S 2 Parallelogram = Base x Height = B x H Triangle = h = Note: Height can drawn and measured outside the figure (parallelogram too). Trapezoid = h = 8 Chapter 8

9 + Find the area of a parallelogram with 10.4 in and height of 3.1 in? A = BxH = (10.4 in)x(3.1 in) A = in 2 A side of a house must be painted. The length = 40 feet and the width = 12 feet. There are two 4 by 4 foot windows (not painted). What is the area to be painted? A = (40)(12) 2(4)(4) A = = 448 ft 2 What is the area of this figure? Figure is triangle. B = 7 m and H = 4 m A = A = 7 4 A =14 m 2 9 Chapter 8

10 What is the area of this Figure? Figure is a trapezoid. A = 11 cm, B = 15 cm, H = 6 cm Area = + A = = 266 A = 78 cm 2 Find the area of the shaded Region? Looks like a rectangle with a triangle cut out of the end. So, area of the rectangle minus the area of the triangle. A R = 15x10 = 150 ft 2 A T = 106=30 ft2 A = = 120 ft 2 Paint the deck. One gallon covers 160 ft 2 and cost \$ What is the cost to paint the deck? Looks like a rectangle with a trapezoid on the end. A R = 31ft x 8 ft = 248 ft 2 A T = 8+166=72 ft2 A = = 320 ft ft 2 x = 2 gallons 2 gal x \$. = \$23.90 Section 8 4 Circles, Circumference, and Area Definitions: Circle all points located the same distance from a fixed point. 10 Chapter 8

11 Center that fixed point. Radius (r) a line from the center to the circle. Diameter (d) a line from the circle through the center on to the circle. 2r = d = Circumference ( C ): - distance around a circle (perimeter). Π = (Pi Greek Letter) if left as the symbol it is exact. Π = (not exact rounded) Π = 3.14 or C = What is the radius? will normally be used. C = 2πR The line is the diameter. d =10.4 m. r = = Chapter 8

12 The line is the radius. r = d = 2( = What is the diameter? Find the circumference? a) exact answer in b) approximate answer using 3.14 C = πd a) C = π10 = 10π ft (exact answer) b) C = 3.14(10) = 31.4 ft (approximate) C = 2πR R = 4.7 cm a) C = 2(π)(4.7 cm) = 9.4π cm b) C = 2(3.14)(4.7 cm) = cm Find the circumference? a) exact answer in b) approximate answer using 3.14 AREA: A = πr 2 = πrxr A = π( )2 = π( ) (rarely used) 12 Chapter 8

13 Find the area of the circle? a) exact b) = (approximate) 21 in = radius A = πr 2 = π(r)(r) a) A = π(21 in)(21 in) = 441π in 2 b) A = = 1386 in 2 Find the area of the circle? a) exact b) = 3.14 (approximate) round to nearest whole unit 9.4 in = diameter R = =4.7 a) A = π(4.7 in)(4.7 in) A =22.09π in 2 b) A=(3.14)(4.7)(4.7)=69.3 in 2 A = 69 in 2 =. A rectangle with two half circles Removed. Find perimeter and Area, use π = 3.14 Perimeter is two staright lines and two half circles. Two half circle = one circle. C = πd A=πRR P = 2(12 cm) + πd P = 24 cm + (3.14)(6) cm P = 24 cm cm = cm Area of rectangle Area Circle A=(12)(6) (3.14)(3)(3) A= = cm 2 Section 8 6 Volume 1 cm 3 = 1 cc = 1ml 1 in 3 (volume has units X 3 ) Cubic 13 Chapter 8

14 Rectangular Solid V = LxWxH Cube V = S 3 = SxSxS Right Circular Cylinder V = πr 2 H Right Circular Cone V = Sphere V = 14 Chapter 8

15 What is the volume of a rectangular Solid 8 in, by 11 in by 2 in? V = L x W x H V = (8 in) x (11 in) x (2 in) V = 176 in 3 Find the volume of a right cylinder with π = 3.14, Radius = 2 in, and Height = 4.5 in? V = πr 2 H V = (3.14)(2 in)(2 in) (4.5 in) V = in 3 Find the volume of a sphere. Use π = 3.14 and Radius = 3 cm. = 4 3 = V = cm 3 Find the volume of a cone. Use π = 3.14, diameter = 8 cm and 18 cm height. R = = = 1 3 =4 V = 3.14(4cm)(4cm)(18cm) V = cm 3 15 Chapter 8

16 a) R = b) R =.. =1.2 =0.8 c) V outer V Inner = (3.14)(1.2)(1.2)(2) (3.14)(0.8)(0.8)(2) V = = mm 3 5 mm 3 16 Chapter 8

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