# Geometry.

## Presentation on theme: "Geometry."— Presentation transcript:

Geometry

Points, Rays, and Lines This is a point. A point represents one place in space. This is a ray. A ray is a straight line of points that begins at one point and goes in one direction. This is a line. A line is a straight path of points that goes in two directions. This is a line segment. A line segment is a straight line that begins and ends on two points. Go On

More Lines This is a horizontal line. This is a vertical line.
These lines are parallel. They run in the same direction and never cross. These lines are perpendicular. They cross at right angles. This is the point of intersection. Go On

Matching Match the symbol on the left with its name on the right.
Line segment Horizontal line Point ray Vertical line Perpendicular lines Parallel lines Line Go On

A degree is a unit which angles are measured.
This is an angle. It is formed by two rays extending from the same point. This is the vertex. It is where the two rays in an angle begin. This is an acute angle. It measures less than This is a right angle. It measures exactly This is an obtuse angle. It measures more than but less than This is a straight angle. It measures exactly Go On This is a reflex angle. It measures greater than

Matching Match the angle on the left with its name on the right.
Reflex angle Right angle Vertex Straight angle Acute angle Obtuse angle Go On

Angles are Everywhere Acute angle Right angle Obtuse angle Go On

More Lines and Angles obtuse acute Parallel lines acute obtuse right
reflex Perpendicular lines Go On

Lines and Angles in Real Life Match the words in the center with their examples on the left or right sides. Perpendicular lines Parallel lines Reflex angle Right angle Obtuse angle Acute angle Go On

Angle Relationships An angle’s name has the vertex as its center. This angle’s name is ABC. This symbol represents an angle. A B C Supplementary Angles Supplementary Angles are two angles that add up to 180°. For example, in the figure below, DEF FEG = 180°. Complementary Angles Complementary Angles are two angles that add up to 90°. For example, in the figure below, WXY YXZ = 90°. W X Y Z D E F G Go On

Adjacent Angles Transversal Line Congruent Vertical Angles Go On
Adjacent angles are two angles that share a side. For example, in the figure below, HIJ and JIL and HIK and KIL are adjacent angles. Transversal Line A Transversal line is a line that crosses parallel lines. In the figure below, the line with points TSPQ is a transversal line. O P Q R S T U V H I J K L Congruent Congruent means equal. When a transversal line cuts through two parallel lines, the alternate interior and alternate exterior angles that are formed are congruent. The following angles are congruent: OPQ RSP QPV PSU TSU SPV RST OPS SPV TSU Vertical Angles The angles that are across from each other (opposite) when two lines intersect each other are called Vertical Angles. Vertical angles (opposite angles) are equal. The following angles are vertical angles: HIK JIL HIJ KIL Go On

Measuring Complementary and Supplementary Angles
When you are using both complementary and supplementary angles, it is important to remember that if you know the measurement of one of the angles, you can find the measurement of the other angle. Remember that Complementary Angles are two angles which add up to 90º. Supplementary angles are two angles which add up to 180º. Click Here for a Problem A B C D 30° Y W X Z 65° Click Here for Answer Go On 60° 115°

Basic Geometric Shapes
Square A Square is made up of 4 right angles (90° angles) and has 4 sides which are equal in length. parallel Rectangle A rectangle is a shape that is made up of 4 right angles. The sides opposite each other are parallel and equal in length. Go On parallel

More Shapes Triangle A Triangle is a shape that has three sides. The 3 angles that make up a triangle equal 180°. Right Triangle A Right Triangle is a shape that has 3 sides. One angle (a right angle) is 90°; the right angle plus the other 2 angles equals 180°. The side opposite the right angle is called the hypotenuse. The other 2 sides are called legs. hypotenuse Legs Circle A Circle is a shape in which all points that make up the circle are exactly the same distance from the center of the circle. The distance across the center of the circle, which cuts the circle in 2 equal parts, is called the diameter. The distance from the center of the circle to a point on the outside of the circle is the radius. diameter radius Go On

Shapes Are All Around Us
Square Circles Triangle Rectangle Right Triangle Go On

The Pythagorean Theorem
Geometry is used in many different occupations by carpenters, architects, graphic artists, and engineers, just to name a few. It is used in designs, construction, and drawings. Many of these workers need to know the Pythagorean Theorem. a b c The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. The formula for finding the length of the sides of any right triangle is as follows: Go On

Using the Pythagorean Theorem
3 4 c Go On

Test Yourself What shape is a floor tile that measures 6 inches by 6 inches? Rectangle Triangle Circle Square 2. If your bedroom measures 15 feet wide and 10 feet long, what is the shape of your bedroom? Square Triangle Circle Rectangle 3. A pizza has a diameter of 12 inches. What shape is your pizza? Square Rectangle Triangle Circle 4. What shape are these objects? Triangles Squares Rectangles Circles 5. A Right Triangle has legs 5 and 12 inches long. How long is the hypotenuse? Go On 17 inches 34 inches 25 inches 13 inches

A Rhombus is also called a Parallelogram.
Polygons Polygons are closed figures (all the segments meet) with three or more sides. Rhombus Octagon Hexagon A Rhombus is also called a Parallelogram. Rectangle Trapezoid Pentagon Diamond Square Triangle Shapes with four sides and only four sides are called quadrilaterals. A Rectangle, Rhombus, Parallelogram, Trapezoid, Diamond, and Square are all quadrilaterals. Go On Triangle

A Quadrilateral shape has ______ sides. One Five Three Four The sides of a Square are Equal Different Long Diagonal A shape with three sides is called a Square Rhombus Circle Triangle How many sides are in a pentagon? One Two Three Five Which of these is not a parallelogram? Rectangle Square Rhombus Hexagon Which of these is not a quadrilateral? Triangle Rectangle Square Rhombus A parallelogram has opposite sides that are Equal Short Long Parallel The distance across the center of a circle is called the Diameter Radius Length Width Go On

Perimeter and Area of Polygons
Perimeter is the distance around the outside of a shape. For example, to find the Perimeter of a Rectangle below, add the lengths of all four sides together. 10 5 The Perimeter of the Rectangle is 30. Area is the measure of the space within the perimeter of a shape. To find the Area of the Rectangle below, multiply one length by one width. 10 5 Go On The Area of the Rectangle is 50.

Area and Perimeter of Triangles
Just like other polygons, to compute the perimeter of a Triangle, you add the sides together. 10 6 The Perimeter of this triangle is 22. To compute the area of a Triangle, you multiply 6 7 10 Go On The Area of this triangle is 35.

Circles The distance around the outside of a circle is called the Circumference. To find the circumference of a circle, you use a symbol called pi ( ). The formula for the circumference of a circle is •diameter. The value of 10 5 diameter radius The circumference of this circle is 31.4. To find the area of a circle, use this formula: (r = radius). Remember, the radius of a circle is half of its diameter. Go On

Practice – Find the Perimeter or Circumference of each shape.
3.5 2 Click for the Problem Click for the Solution 10 4 3 5 Go On

Find the Area of each Shape.
11 6 3 Click for the Problem Click for the Solution 6 8 6 Go On

Three-Dimensional Solids
Cube Sphere Rectangular Solid Cylinder Cone Go On

Three-Dimensional Solids
Cube Sphere Cone Rectangular Solid Go On Cylinder

Volume Volume is the measure of space inside a three-dimensional (not flat) figure. For a cube or rectangular solid, use the formula Cube Rectangular Solid For a cylinder, use this formula: Go On Cylinder

To find the volume of a sphere, use this formula:
Volume Formulas To find the volume of a cone, use this formula: Cone To find the volume of a sphere, use this formula: Go On Sphere

A box is 10 inches long by 5 inches wide by 7 inches tall. What is its volume? 10 5 7 8 5 A soccer cone is 8 inches high, with a radius of 5 inches. What is its volume? Go On

Practice A pair of fuzzy dice have sides which are 4 inches each. What is the volume of one die? 4 20 10 4 4 What is the volume of a can of soup that has a radius of 10 cm and a height of 20 cm? Go On

Review Horizontal Line Acute Angle Vertical Line Right Angle
Parallel Lines Obtuse Angle Perpendicular Lines Straight Angle Line segment Reflex Angle Go On Ray

Review Supplementary Angles add up to 180°
Complementary Angles add up to 90° Adjacent Angles are two angles that share a side. A Transversal Line is a line that crosses parallel lines. Vertical Angles (Opposite Angles) are Congruent. Congruent means equal. Go On

Review Pentagon A Polygon is a figure with three or more sides.
A Quadrilateral Polygon has four sides. The Pythagorean Theorem is Octagon Hexagon Rhombus Legs hypotenuse Rectangle Right Triangle Trapezoid Triangle diameter radius Square Diamond Go On Circle (A Circle is not a polygon.)