II. Geometry and Measurement


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1 II. Geometry and Measurement The Praxis II Middle School Content Examination emphasizes your ability to apply mathematical procedures and algorithms to solve a variety of problems that span multiple mathematics content areas. Two related content areas are geometry and measurement. According to the PRAXIS II Study Guide, these content areas account for 25% of the mathematics component of the Middle School Content Examination and include the following topics. Geometry (20%): Examinees should have knowledge of relationships in both two and three dimensions, as well as draw inferences on the concepts of parallelism, perpendicularity, congruence and similarity, angle measure, and polygons. Measurement (5%): Examinees should have knowledge and application of standard units of both the English and metric systems, nonstandard units, estimation, perimeter, area, volume, mass, weight, angle measure, time and temperature. Geometrical shapes and solids are all around us. Look out the window and you may see angles, triangles, squares, or circles. In ancient times, people built and interacted with a variety of objects and structures. The shapes and solids that recurred most often were named and their properties were discovered. In this section, you will explore the properties of these twodimensional shapes.
2 Topic A: Geometry A1. Triangles Triangles come in many shapes and sizes. In nature, we often see triangles in support structures for buildings and bridges. Triangles can be classified by their sides or their angles. A triangle can have no equal sides, two equal sides, or three equal sides. It can also have three acute angles (angles that are less than 90 ), one right angle, or one obtuse angle (angles that are greater than 90 ). The names of these triangles are shown in the table below. Scalene Triangle Isosceles Triangle Equilateral Triangle The sides of a scalene triangle are all different lengths. Scalene triangles can be acute, obtuse, or right triangles. Obtuse Triangle An isosceles triangle has two sides that are the same length and two interior angles that are the same measure. Isosceles triangles can be acute, obtuse, or right triangles. Right Triangle All three sides of an equilateral triangle are the same length and all interior angles measure 60. All equilateral triangles are acute triangles. Acute Triangle Exactly one of the interior angles of an obtuse triangle is obtuse (an obtuse angle is over 90 degrees). Obtuse triangles can be scalene or isosceles triangles A right triangle has one angle that is a right angle (as marked above). In a right triangle the side opposite the right angle is the longest side and called the hypotenuse. Right triangles can be scalene or isosceles triangles. All three interior angles of an acute triangle are acute (less than 90 degrees). All equilateral triangles are acute triangles and some isosceles and scalene triangles are also acute triangles.
3 A Venn Diagram is a graphical organizer that shows the relationships between groups of things. Using the definitions above, we can create a Venn Diagram to so the relation between triangles. This diagram shows that all equilateral triangles are acute triangles. No equilateral triangle can be a scalene triangle. Some acute triangles are scalene triangles and vice versa. Acute Triangles Equilateral Triangles Scalene Triangles There is one important property of triangle that deals with the interior angles of a triangle and one property that deals with the sides of the triangle. Property 1: The sum of the interior angles of a triangle sum to 180. Property 2: The sum of any two sides of a triangle must be greater than the third side > > > 4 To practice Property 2, please visit the Triangle Constructed Response or Lengths of Sides of Triangle Learning Objects.
4 The Pythagorean Theorem, dealing with the relationships in right triangles, is one of the most famous theorems in geometry. This theorem describes the relationship between the lengths of legs of a right triangle and the hypotenuse. Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse. When the lengths of the legs are a and b, and the length of the hypotenuse is c, then a 2 + b 2 = c 2. In the following learning objects, you will apply Pythagorean s Theorem to solve problems. Cross Sectional Area Coordinate Geometry A2. Quadrilaterals The word quadrilateral is derived from two Latin words quadri which means four and later which means sides. Therefore, a quadrilateral is a polygon with four sides. There are many different quadrilaterals that have important geometric properties as described in the table below. A trapezoid is a quadrilateral that has exactly one pair of parallel sides. An isosceles trapezoid is a trapezoid (in yellow) in which the nonparallel sides are congruent (same length). A kite is a quadrilateral that has two pair of adjacent sides that are congruent (same length).
5 A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. This means that opposite sides are also congruent (or the same length). In a parallelogram, the opposite angles are congruent (or the same measure) and the adjacent angles are called supplementary angles because they add to 180 degrees. A rectangle is a parallelogram that has four right angles. Since a rectangle is a parallelogram it must have opposite sides that are parallel and congruent. A rhombus is a parallelogram that has four congruent sides. Since a rhombus is a parallelogram it must have opposite sides that are parallel and congruent. A square is a rectangle that has four congruent sides. A square is a rhombus that has four right angles. A square is equilateral (equal side lengths) and equiangular (equal angle measures). The Venn Diagram below shows the relationship between these special quadrilaterals. Rectangles Trapezoids Squares Rhombuses Parallelograms Below are seven statements about quadrilaterals. Can you identify which ones are always true, sometimes true, or never true?
6 1. A trapezoid is a parallelogram. 2. A square is always a rectangle. 3. A rectangle has four right angles. 4. A rhombus always has four equal sides. 5. A parallelogram has one set of opposite sides. 6. A parallelogram is never a square. 7. A rhombus is never a square. To visualize other geometric relationships, visit the Venn Diagram Learning Object. There is one important property of quadrilaterals that deals with the interior angles. The sum of the interior angles in any quadrilateral adds to 360 degrees. Other properties of quadrilaterals can be summarized in the following table. The last two rows provide facts about the diagonals of quadrilaterals. The diagonals are line segments drawn from one vertex to the opposite vertex as shown in the parallelogram to the right. Properties of Isosceles Parallelogram Rhombus Rectangle Square Quadrilaterals Trapezoid Trapezoid Kite Exactly 1 pair of Never Never Never Never Always Always Never parallel sides 2 pairs of parallel sides Always Always Always Always Never Never Never Exactly 1 pair of Never Never Always Never congruent sides 2 pairs of congruent Always Always Always Always Never Never Always sides All sides congruent Always Always Never Never Never
7 Opposite angles Always Always Always Always Never Never Always congruent Adjacent angles are Always Always Always Always Never Never Always supplementary (add to 180 degrees) Perpendicular Always Always Never Never Always diagonals Congruent diagonals Always Always Always Never A3. Other Polygons A polygon is a many sided figure that has an equal number of sides as angles. Triangles and quadrilaterals are two examples of polygons. Other polygons are name according to the number of sides (or number of angles) that they have. The picture to the right is a pentagon because it has five sides. A polygon that has congruent angles and congruent sides is called a regular polygon. A regular polygon is equiangular and equilateral. Polygon Names Number of Sides Name Number of Sides Name 3 Triangle 7 Heptagon 4 Quadrilateral 8 Octagon 5 Pentagon 9 Nonagon 6 Hexagon 10 Decagon In a triangle the sum of the interior angles is 180. In a rectangle the sum of the four interior angles is 360. This fact can be extended to all quadrilaterals. The process below to verify this property can be extended to all polygons. The sum of the interior angles of a quadrilateral is equal to 360. In this quadrilateral we can draw a diagonal from one vertex to the opposite vertex creating two triangles. The sum of the interior angles in each of the triangles is 180. This makes the sum of the interior angles of a quadrilateral equal two = 360
8 The sum of the interior angles of a quadrilateral is equal to 540. In this pentagon we can draw two diagonals from one vertex to the opposite vertices creating three triangles. The sum of the interior angles in each of the triangles is 180. This makes the sum of the interior angles of a pentagon equal two = 540 In the learning object, Polygon Sum Conjecture, you will examine how to find the sum of the interior angles for any polygon. A4. Symmetry Symmetry adds beauty and balance to natural forms and architectural design. The Wisconsin Model Academic Standards for Mathematics Glossary ( terms defines symmetry as follows. A figure has symmetry if it has parts that correspond with each other in terms of size, form, and arrangement. There are two common types of symmetry shown in the pictures to the left. The first column represents reflection (line) symmetry and the second column represents rotational symmetry. A figure with reflection symmetry has two halves which match each other perfectly if the figure is folded along its line. This type of symmetry is also called line or mirror symmetry. A plane figure may have more than one line of symmetry as shown below.
9 A figure has rotation symmetry or turn symmetry if the figure comes back to itself when rotated through a certain angle between 0 and 360. Many shapes have rotational symmetry. For example, a rectangle can be turned 180 and it will come back to itself. An equilateral triangle has two rotation symmetries at 120 and 240. A figure with a 90 rotation (like the leafs and flowers below) will also have 180 and 270. A5. Transformations Two polygons are congruent if they have the same shape and the same size. In geometry, two polygons are congruent if their corresponding sides are equal in length and their corresponding angles are equal in measure. As an example, the two triangles below are congruent. Δ ABC ΔDEF If you ask a child to determine if two polygons are congruent, they may place one shape over the other. Consider the types of moves a child would have to make to see that the triangles below are congruent.
10 The set of rigid transformations consists of the various ways to move a geometric figure around while preserving the distance between the points in the figure. Rigid transformations always produce congruent figures. In the learning object Reflection, you will explore these geometrical transformations. A6. Coordinate Geometry Coordinate Geometry The coordinate plane is a basic concept for coordinate geometry. The coordinate plane is also called the xyplane and it is useful for plotting points, lines, and figure. In the coordinate plane to the right, point L is represented by the coordinates ( 3, 1.5) because it is positioned on 3 along the xaxis and on 1.5 along the yaxis. Similarly, you can figure out why the points M = (2, 1.5) and N = ( 2, 3). In coordinate geometry, we often use the xy plane to find the distance between two points. As an example, what is the distance between points M and N. To do this we can construct a right triangle where the line segment joining points M and N serves has the hypotenuse of this triangle.
11 The distance between the two points is the length of the hypotenuse which can be found using Pythagorean s Theorem. c 2 = c 2 = c = units For more information, visit the following learning object. Coordinate Geometry Points on a Line Topic B Measurement B1.Standard Units of Measurement What does it mean to measure something? Measurement is the assignment of a numerical value to an attribute of an object, such as the length of a pencil, the area of a playground, the weight of a rock, or the temperature of a hot cup of coffee. The units that we use to measure are most often standard units, which means that they are universally available and are the same size for all who use them. Sometimes we measure using nonstandard units, which means that we are using units that we have invented. For example, we could measure the length of a room in arm spans instead of meters. Two standard measurement systems have been developed and universally agreed upon. These systems of measurement are called the English System and the Metric System. The most frequently used units of measurement in the English System are provided below. Length: Foot Capacity/Volume: Temperature: Weight/Mass: Pound Gallon Fahrenheit 1 yard = 3 feet 1 mile = 5,280 feet 12 inches = 1 foot 1 pint = 2 cups 1 quart = 2 pints 1 gallon = 4 quarts 16 ounces = 1 pound The Metric System resembles our decimal system and is based on powers of ten. Conversions are made by moving the decimal place to the left or right. The standard unit of length in the metric system is the meter. Other units of length such as the kilometer and the centimeter can be found by multiplying by the appropriate power of 10 as described in the table below.
12 kilo thousands hecto 100 hundreds deca 10 tens unit deci 0.1 tenths centi hundredths milli thousandths 1000 meters = 1 kilometer (km) 100 meters = 1 hectometer 10 meters = 1 decameter 1 meter = 10 decimeters 1 meter = 100 centimeters (cm) 1 meter = 1000 millimeters (mm) In the metric system, the basic unit for weight/mass is the gram. The basic unit for capacity/volume is liters. The basic unit for temperature is Celsius. Take the short Metric Mayhem Quiz to test your ability to compare metric measurements or visit the Units of Conversion learning object. There are standard tables to convert from the English system to the metric system. Here are some examples. This paperclip has a length of 1 inch and weighs 1 gram.. We could also say that this paperclip has a length of about 2.54 centimeters and weighs about ounces. The ruler below is 1 foot long. Look closely because you will also see that it is about 30 centimeters long. Actually 1 foot = centimeters = meters. A gallon of water is equivalent to liters. The optimal temperature for a bath is 38 degrees Celsius. This is equivalent to 100 degrees Fahrenheit. The typical temperature conversion formula is given by F = 9/5C Unit Analysis Unit analysis or dimensional analysis is a procedure fro converting from one unit of measurement to another unit of measurement. It uses proportional reasoning to transform form one unit to the next without having to know all the standard conversion factors. For example, how many meters are in the 400yard dash? Since we do not know how many meters in a yard, we will use the fact that 1 foot = meters to solve this problem. 3 feet meters 400 yards = = 76 1 yard 1 foot ( ) meters 365. meters
13 If Mai finishes the 400yard dash in 8 minutes, then how fast did she run in miles per hour? To answer this question we will use the fact that 5,280 feet = 1,760 yards = 1mile. 400 yards 4min 1mile 60min 17760yards 1hour = ( ) ( ) miles per hour 3.41miles per hour To learn more about unit analysis, visit the following learning object. Units of Measure Unit Analysis Proportional Reasoning B2. Area and Perimeter Area and Perimeter The area of a shape is typically defined as the number of square units that are required to completely cover that shape. In the picture below, the area of the rectangle is 35 square units. Common units of measurement for area are square centimeters (cm 2 ) or square feet (ft 2 ). The perimeter is a measure of length so it is given the units centimeters or feet. The perimeter can be translated as the around measure or the length of the closed curve. For a polygon, the perimeter is the sum of the lengths of all its sides. For a circle, the perimeter is often referred to as the circumference. The most common two dimensional figures for which areas and perimeters must be found are triangles, parallelograms, rectangles, squares, trapezoids, and circles. This section will develop a formula for finding the area of these polygons.
14 Rectangles Squares Height = h Area = s 2 Perimeter = 4s Area = bh Perimeter 2b + 2h Parallelogram Base = b Triangle Side = s Area = (area of rectangle) = bh Perimeter: Add up lenghts of 4 sides. Circle Area = ½(area of parallelogram) = 1/2bh Perimeter: Add up the lengths of the 3 sides Trapezoid Area = πr 2 Circumference = 2πr = πd Area = (1/2 area of parallelogram) = ½(a + b)h Perimeter: Add up the lengths of the 4 side
15 Visit the following learning objects to practice your skills using these formulas. Area Learning Object Dimensions Cross Section Area To learn more about circles, visit the Properties of Circles Leaning Object. B3. Similar Figures In geometry, two polygons are similar if they have the same shape (not necessarily the same size). The picture below gives examples of similar polygons. Triangles are similar if their corresponding (matching) angles are equal and the ratio of their corresponding sides are in proportion. Δ ABC ~ ΔDEF Corresponding Sides are in Proportion 6 12 = 8 16 = = 1 2 Quadrilaterals are similar if their corresponding (matching) angles are equal and the ratio of their corresponding sides are in proportion. Corresponding Sides are in Proportion = = 6 9 3
16 Similar Triangles: Corresponding sides of the similar triangles below are proportional since = = =. What is the ratio of the triangles perimeters? The ratio of the perimeters is also proportional to the ratio of corresponding sides, 24 1 =. In other words, if two 48 2 polygons are similar, their corresponding sides, heights, diagonals, and perimeters are all in the same ratio. Perimeter = = 24 units Perimeter = = 48 units Similar Rectangles: Corresponding sides of the similar rectangles below are proportional since = =. What is the ratio of the areas? The ratio of the areas is given by = =. In other words, if two polygons are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Area: 9 x 6 = 54 square units Area: 3 x 2 = 6 square units To work with similar polygons, visit the Ratio of Inscribed Square learning objects.
17 B4. Surface Area and Volume Threedimensional figures or space figures can also be found right outside your window. Many every day objects, such as boxes, cans, wedges, and spheres, represent special geometrical solids. If a solid is made up of polygonal faces, then we call this solid a polyhedron. For example a box and wedge are polyhedra. The correct mathematical term to describe a polyhedron is to name it according to the number of faces. A box has six faces so we refer it to as a hexahedron. The wedge is a pentahedron with five faces. Cones, cylinders, and spheres are special types of solids that are not examples of polyhedra. The cereal box and the cheese wedge are common prisms. A prism has two bases that are congruent polygonal regions that lie in parallel planes. The lateral faces that join the base are all parallelograms. A box is also referred to as a rectangular prism and it has 8 vertices and 12 edges. The wedge is referred to as a triangular prism with 6 vertices and nine edges. Other special polyhedrons can also be found in nature and architecture. The Great Pyramid of Giza represents a pentahedron since it has a square base and four triangular faces. In geometry, a pyramid has triangular faces which meet a common point called the apex. The base of a pyramid can be any polygonal region which is used to name the pyramid. Polyhedra can be constructed from twodimensional shapes called nets that can be folded into the appropriate form. Can you name which net belongs to the given polyhedra?
18 How much will a container cost to manufacture? How much paint do you need to paint a bedroom? To answer these questions you must find the surface area. The total surface area is the sum of the area of all of its surfaces. A surface area in square units indicates how many squares it would take to wrap the outside of the space figure. The surface area is also the combined twodimensional area of a geometrical net. There are many formulas to find surface are as shown in the table below. Solid Net Formula Rectangular Prism 2 x area of rectangular base + 4 x area of rectangular sides. Square Pyramid Area of square base + 4 x area of triangular sides.
19 Cylinder 2 x are of circular bases + area of rectangular side Volume is the amount of space occupied by a threedimensional figure. Volume is typically measured in cubic units, but may also be described as gallons, fluid ounces, or even the number of people required to fill a phone booth. Whereas, surface area is the total areal of the faces of the solid, volume is the capacity of the solid. In other words, surface area can be thought of as wrapping paper, volume can be considered as the filling. As with surface area, there are many known formulas for calculating the volume of prisms, cylinders, pyramids, and other space figures. Solid Examples Formula Generalization Rectangular Prism Area of the rectangular Base B = l x w Triangular Prism Cylinder Volume V = l x w x h = Bh where B is the area of the base. Area of triangular base B = ½bh Volume V = Bh where B = area of the base. Area of the circular Base B = πr 2. Volume V = πr 2 h = Bh where B is that area of the base. Prisms and Cylinders V = Bh where B = area of the base
20 Square Pyramid Area of square Base B = b 2 Volume V = 1/3b 2 h = 1/3Bh where B is the area of the base. Pyramids Volume V = 1/3Bh where B is the area of the base.
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