Tennessee Adult Education Mathematics Curriculum 2011 Pre-GED Geometry Lesson 7

What is Geometry? It is the branch of mathematics that deals with lines, points, curves, angles, surfaces, and solids.

Lines - Basic Terms Term Definition Point (•) A location on an object or a position in space. Line A connected set of points that extends without end in two directions. Line Segment A piece of a line, like a jump rope, that is a specific length. Ray Part of a line that extends indefinitely in one direction.

Naming Lines Lines are named by the points that are located on them. Example: This line can be named in 2 ways: AB or BA A B This reads “line AB” Line segments are named by the points that are located on them. Example: This line segment can be named in 2 ways: CD or DC C D This reads “line segment CD”

Naming Rays Rays are named by the points that are located on them. Example: F E EF There is only one way to name a ray. The end point must go first! This reads “ray EF”

Guided Practice Directions: Name the figures. 2. _______ 1. _______ C B D A E 3. _______ F

Guided Practice Directions: Name the figures. CD or DC AB BA 2. _______ 1. _______ or C B D A E EF 3. _______ F

Lines that form a right angle when they intersect Term Definition Parallel Lines Lines that are always the same distance apart from each other. They will never intersect. Perpendicular Lines Lines that form a right angle when they intersect Intersecting Lines Lines that cross, or that will cross. The point at which they cross is called the vertex. Transversal Lines is a line that intersects a set of parallel lines.

Naming Parallel Lines When naming parallel lines, first name the lines, then place the symbol for parallel in the middle. Symbol for parallel: II R S RS II TU T U This reads “ line RS is parallel to line TU”

Naming Perpendicular Lines When naming perpendicular lines, first name the lines, then place the symbol for perpendicular in the middle. Symbol for perpendicular: R RS TU T U S This reads “ line RS is perpendicular to line TU”

Guided Practice Directions: Tell whether each figure is perpendicular or parallel, then name the lines. U Q S W X R T V 1. _____________________ 2. _____________________

Guided Practice Directions: Tell whether each figure is perpendicular or parallel, then name the lines. U Q S W X R T V parallel QR II ST perpendicular UV WX 1. _____________________ 2. _____________________

Basic Shapes Polygons Quadrilateral Are closed flat shapes with 3 or more straight sides. Quadrilateral are polygons that have four sides and four angles. Name Shape Triangles Parallelograms Quadrilaterals Rectangles Pentagon Square Hexagon Rhombus Octagon Trapezoid

quadrilaterals A four sided figure a b Parallelogram: has 2 sets of parallel sides. c d AB ll CD and AC ll BD Rectangle: is a parallelogram, each set of parallel sides must be equal in length, and must form 4 right angles. The small boxes in the corners indicate a right angle. Angles will be discussed later in the lesson

More quadrilaterals 4 inches Square: is a parallelogram, all sides are equal in length, and must have 4 right angles 4 inches 4 inches 4 inches 5 inches 5 inches Rhombus: is a parallelogram, all sides are equal in length, angle measurement does not matter. 5 inches 5 inches A B Trapezoid: has one set of parallel sides AB II CD C D AC is not parallel to BD

Guided Practice Directions: Name the Quadrilaterals. 1. ________________ 2. ________________ 3. _________________ _________________ 2. 3. 4. 5.

Guided Practice Directions: Name the Quadrilaterals. 1. Trapezoid ________________ 2. ________________ 3. _________________ _________________ Parallelogram 2. Square 3. Rectangle Rhombus 4. 5.

What are angles? An angle measures the amount of turn. As the Angle Increases, the Name Changes. Pictures from clipart

Type of Angle Description Acute An angle less than 90° Right An angle that is 90° exactly Obtuse An angle that is more than 90° Straight An angle that is exactly 180° Reflex An angle that is greater than 180°

Parts of an Angle The point at which the two rays meet is called the vertex. The two straight sides are called rays. The angle is the amount of turn between each ray. Ray Ray angle · Vertex

Guided Practice Directions: Classify each angle as acute, right, obtuse, or straight. 1. ___________ 2. ___________ 3. ___________ 4. ___________ 5. ___________ 6. ___________

Guided Practice Directions: Classify each angle as acute, right, obtuse, or straight. Obtuse 1. ___________ 2. ___________ Right Acute 3. ___________ Right Acute 4. ___________ 5. ___________ 6. ___________ Straight

Naming Angles There are three ways to name angles: Name an angle by the vertex. For example: B Name an angle by all three letters. For example: ABC CBA HINT: The vertex is always the middle letter

Guided Practice Directions: Name each angle in three ways. _____ G I L J K H M _____ N O

Guided Practice Directions: Name each angle in three ways. JKL _____ LKJ G I GHI K _____ L IHG J K H H M MNO _____ N ONM O N

Supplementary Angles The two angles below (140⁰ and 40⁰) are supplementary angles, because their measurements add up to 180⁰. NOTICE: When the two angles are put together, they form a straight line.

Complementary Angles The two angles at the right (40° and 50°) are Complementary Angles, because they add up to 90°. NOTICE: When the two angles are placed together, they form a corner.

Complementary vs Supplementary How can you remember which is which? Easy! Think: "C" of Complementary stands for "Corner" (a Right Angle), and "S" of Supplementary stands for "Straight" (180 degrees is a straight line)

Guided Practice Directions: Find the missing angles using complementary or supplementary rules. 1. 2. 160˚ x x 52˚ x = _________ x = _________ 3. 4. x 43˚ x 100˚ x = _________ x = _________

Guided Practice Directions: Find the missing angles using complementary or supplementary rules. 1. 2. 90 - 52 160˚ 180 - 160 x x 52˚ 38 20˚ 20 x = _________ x = _________ 38˚ 3. 4. 90 - 43 180 - 100 x 43˚ x 100˚ 47 80 47˚ x = _________ x = _________ 80˚

a + b + c = 180⁰ What are Triangles? A triangle has three sides and three angles The three angles always add to 180° a a + b + c = 180⁰ b c

Triangle Example Add the two known angles, then subtract the total from 180. x Step 1: Step 2: 80 + 80 = 160 180 - 160 20 x = 20˚ 80˚ 80˚ Check: 80 + 80 + 20 = 180

Guided Practice Directions: Find the missing angle measurement for each triangle. 1. 60˚ x = _________ x = _________ 60˚ x x 3. 20˚ 20˚ 2. 20˚ x = _________ x 80˚

Guided Practice Directions: Find the missing angle measurement for each triangle. 1. 20 + 80 = 100 60˚ 60˚ x = _________ 60 + 60 = 120 180 - 100 180 - 120 80˚ 80 x = _________ 60˚ x 60 x 3. 20˚ 20˚ 2. 20˚ 140˚ x = _________ 20 + 20 = 40 180 - 40 140 x 80˚

Area of a Triangle "b" is the distance along the base of the triangle "h" is the height (measured at right angles to the base) Area = base x height 2

Example: What is the area of this triangle?   ft. Hint: The fraction line means divide. Therefore, 240 ÷ 2 = 120 ft. Every area is expressed in square units

Guided Practice Find the area of each triangle Remember to put the answer in squared units 2. 1. 8 feet 4 feet 7 feet 5 feet 3. 10 inches 2 inches

Guided Practice Find the area of each triangle Remember to put the answer in squared units. A= b x h 2 A= b x h 2 A = 5 x 8 2 2. 1. A = 4 x 7 2 A = 40 2 8 feet 4 feet A = 28 2     7 feet 5 feet 3. A= b x h 2 A = 2 x 10 2 10 inches A = 20 2   2 inches

Finding the Perimeter 24 in. 10 in. 24 + 24 + 10 + 10 = 68 in. The perimeter is the distance around a figure such as a square, a rectangle, or a triangle. To find the perimeter of a figure, you simply add up all the sides. Example: 24 in. 10 in. 24 + 24 + 10 + 10 = 68 in.

Let’s Practice How many feet of fencing material will Louise need to surround her garden that measure 20 feet on two sides and 10 feet on the other two sides?

Let’s Practice How many feet of fencing material will Louise need to surround her garden that measure 20 feet on two sides and 10 feet on the other two sides? 20 + 20 + 10 + 10 = 60 Feet

More Practice Remember to add ALL sides 1. 5 in. 2. 2 cm. 15 cm. 23 mm. 4. 6 ft. 3. 18 mm. 10 ft. 8 ft.

More Practice Remember to add ALL sides 1. 15 cm. 5 in. 2. 2 cm. 2 cm. 15 + 15 + 2 + 2 = 34 cm. 5 + 5 + 5 + 5 = 20 in. 23 mm. 4. 6 ft. 6 ft. 3. 18 mm. 18 mm. 10 ft. 10 ft. 23 mm. 23 + 23 + 18 + 18 = 82 mm. 8 ft. 6 + 6 + 8 + 10 + 10 = 40 ft.

More Geometry operations and terms! These will be discussed more in-depth in the next level. 1. Circumference of a circle 2. Area of a circle 3. Transversal Lines 4. X,Y Coordinates

Circumference: the perimeter around a circle. Key terms: Diameter – is the distance across the circle. The symbol for diameter is “d”. Radius – All the points on the outside of the circle are equal distances from the center of a circle. The symbol for the radius is “r”. Note: Diameter = 2(Radius) Or D=2R

How to find the circumference:   28 in

Area of a circle   HINT: 6² = 6 x 6 6 in ²

Angles Formed by A Transversal A transversal line is a line that cuts through a set of parallel lines. As the transversal cuts through, it forms both Corresponding and Vertical Angles AB CD This reads as Line AB is parallel to Line CD. Corresponding angles have equal measurements, and vertical angles have equal measurements. Transversal line

X, y charts Vocabulary X axis = the horizontal axis ( ) Coordinate Plane Vocabulary X axis = the horizontal axis ( ) Y axis = the vertical axis ( ) Coordinates = are a set of points that give a specific location on a map or a chart. (x, y) For example: ( 2, 4) The first number is the x. Begin at 0. Since the number is positive move to the right to the 2. The second number is the y. Start from the 2. Since the number is positive, move up 4. 5 4 (2 , 4) 3 2 -4 1 -5 -3 -2 -1 x 1 2 3 4 5 -1 -2 -3 -4 -5 y