Math 6, Unit 8 Notes: Geometric Relationships Points, Lines and Planes; Line Segments and Rays As we begin any new topic, we have to familiarize ourselves with the language and notation to be successful. My guess is that you might already be pretty familiar with many of the terms about to be introduced in this section; the biggest difference is that we will formalize our understanding and introduce notation that will enable us to express that knowledge quickly. Let s look at one of our first elements in geometry, a point. A point is pictured by a dot. While a dot must have some size, the point it represents has no size. Points are named by capital letters. P This point would be read point P. A line extends indefinitely. A line, containing infinitely many points, is considered to be a set of points, hence it has no thickness. A line can be named by a lower case letter or by two points contained in the line. R S k This line could be called line k or RS, read line RS. Note that RS does not begin or end at either of the points R or S. A plane is a flat surface. Such things as table tops, desks, windowpanes, and walls suggest planes. A plane, like the aforementioned, does not have thickness and extends indefinitely. C B A A plane is named by 3 points that are on the plane (but not the same line called noncollinear points). This plane could be called plane ABC, or plane CAB or plane ACB. A line segment contains two endpoints and all the points between those endpoints. B A A line segment is named by its endpoints. The above example could be read line segment AB or AB, which is also read line segment AB. A ray, denoted by XY, has one endpoint from that endpoint, the ray extends without end (in one direction). Note that the endpoint is named first, telling you that in this case the ray begins at point X and passes through point Y. Math 6 Notes Unit 8: Geometric Relationships Page 1 of 13 Revised 013 CCSS
This could be called ray XY and shown by XY. Y X An angle is formed by the union of two rays with a common endpoint, called the vertex. Angles can be named by the vertex. V This angle would be called angle V, shown as V. However, the best way to describe an angle is with 3 points: one point on each ray and the vertex. When naming an angle, the vertex point is always in the middle. S N U This angle can now be named three different ways: SUN, NUS, or U. You can classify an angle by its measure. Acute angles are greater than 0, but less than 90º. In other words, not quite a quarter rotation. Right angles are angles whose measure is 90º. Obtuse angles are greater than 90º, but less than 180º. That s more than a quarter rotation, but less than a half turn. And finally, straight angles measure 180º. acute right obtuse straight Angle Relationships Syllabus Objectives: (6.1)The student will model the measures of complementary and supplementary angles with and without tools of measurement. Syllabus Objectives: (6.13)The student will find the measures of complementary and supplementary angles with and without tools of measurement. Vertical angles are formed when two lines intersect they are opposite each other. These angles always have the same measure. We call angles with the same measure congruent. P T Q S R PQR and TQS are vertical angles. PQT and RQS are vertical angles. Math 6 Notes Unit 8: Geometric Relationships Page of 13 Revised 013 CCSS
Adjacent angles are two angles that have a common vertex, a common side (ray), and no common interior points. P T Q S R PQR and PQT are adjacent angles. PQT and TQS are adjacent angles. TQS and SQR are adjacent angles. SQR and RQP are adjacent angles. We call two angles whose sum is 90º complementary angles. For instance, if m X = 40 and m Y = 50, then X and Y are complementary angles. If m A= 30, then the complement of A measures 60. Two angles whose sum is 180º are called supplementary angles. If m M = 100 and m S = 80, then M and S are supplementary angles. Example: 4 3 1 1. What angle is vertical to 1?. Name an angle adjacent to 4. 3. If m 3 = 50, what are the measures of 1,, and 4? 4. What is the sum of the measures of 1,, 3, and 4? 1. 3 is vertical with 1. Either 3 or 1 is correct 3. m 1 = 50 because vertical angles are congruent m = 130 because and 3 form a straight angle (180 50 ) m 4 = 130 because it is vertical with and therefore congruent OR because 3 and 4 form a straight angle (180 50 ) 4. 360 Math 6 Notes Unit 8: Geometric Relationships Page 3 of 13 Revised 013 CCSS
CRT Example: CRT Example: Classifying Lines Two lines are parallel lines if they do not intersect and lie in the same plane. The symbol is used to show two lines are parallel. Triangles ( ) or arrowheads (>) are used in a diagram to indicate lines are parallel. l m l m Two lines are perpendicular lines if they intersect to form a right angle. The symbol is used to state that two lines are perpendicular. p q p f g f g > > q Math 6 Notes Unit 8: Geometric Relationships Page 4 of 13 Revised 013 CCSS
Two lines are skew lines if they do not lie in the same plane and do not intersect. r t Lines r and t are skew lines. Triangles Triangles can be classified by the measures of their angles: acute triangle 3 acute angles right triangle 1 right angle obtuse triangle 1 obtuse angle Triangles can also be classified by the lengths of their sides. You can show tick marks to show congruent sides. equilateral triangle 3 congruent sides isosceles triangle at least congruent sides scalene triangle no congruent sides equilateral isosceles scalene and and and acute acute acute scalene isosceles scalene and and and right obtuse obtuse Math 6 Notes Unit 8: Geometric Relationships Page 5 of 13 Revised 013 CCSS
Syllabus Objective: (5.4) The student will derive formulas for finding the areas of plane figures. Syllabus Objective: (5.5) The student will model formulas to find the perimeter, circumference, and area of plane figures. NOTE: Syllabus objective 5.5 (above) and 5.6 (The student will apply formulas to find the perimeter, circumference, and area of plane figures) appear in the 4 th quarter. Therefore we took the liberty to interpret this to mean that students would informally work with perimeter, circumference and area of plane figures in this chapter. Areas of Plane Figures Rectangles/Squares/Rhombi One way to describe the size of a room is by naming its dimensions. A room that measures 1 ft. by 10 ft. would be described by saying it s a 1 by 10 foot room. That s easy enough. There is nothing wrong with that description. In geometry, rather than talking about a room, we might talk about the size of a rectangular region. For instance, let s say I have a closet with dimensions feet by 6 feet. That s the size of the closet. ft. 6 ft. Someone else might choose to describe the closet by determining how many one foot by one foot tiles it would take to cover the floor. To demonstrate, let me divide that closet into one foot squares. ft. 6 ft. By simply counting the number of squares that fit inside that region, we find there are 1 squares, or 1 square foot. If I continue making rectangles of different dimensions, I would be able to describe their size by those dimensions, or I could mark off units and determine how many equally sized squares can be made. Rather than describing the rectangle by its dimensions or counting the number of squares to determine its size, we could multiply its dimensions together. Putting this into perspective, we see the number of squares that fits inside a rectangular region is referred to as the area. A shortcut to determine that number of squares is to multiply the by the. More formally area is defined as the space inside a figure or the amount of surface a figure covers. The Area of a rectangle is equal to the product of the length of the and the length of a to that. That is A = bh. Most books refer to the longer side of a rectangle as the length (l), the shorter side as the width (w). That results in the formula A = lw. So now we have formulas for the areas of a rectangle that can be used Math 6 Notes Unit 8: Geometric Relationships Page 6 of 13 Revised 013 CCSS
interchangeably. The answer in an area problem is always given in square units because we are determining how many squares fit inside the region. Of course you will show a variety of rectangles to your students and practice identifying the and of those various rectangles. If we were to have a square whose sides measure 5 inches, we could find the area of the square by putting it on a grid and counting the squares as shown below. Again we could count the boxes to find the area of the square ( 5units ), but more easily we could multiply the x or the length x the width ( 5x5 = 5units ). So again, A=bh or A = lw. Students may also see that since the and or length and width of a square are congruent, they may choose to use A= s ( 5x5 = 5units ). Next, if we begin with a 6 x 6 square and cut from one corner to the other side (as show in light blue) and translate that triangle to the right, it forms a new quadrilateral called a rhombus (plural they are called rhombi). Since the area of the original square was 6 x 6 or 36 square units then the rhombus has the same area, since the parts were just rearranged. So we know the rhombus has an area of 36 square units. Again we find that the Area of the rhombus = bh. Be sure to practice with a number of different rhombi so your students are comfortable with identifying the versus the width of the figures. Math 6 Notes Unit 8: Geometric Relationships Page 7 of 13 Revised 013 CCSS
Areas of Plane Figures Parallelograms If I were to cut through one corner of a rectangle and place it on the other side, I would have the following: We now have a parallelogram. Notice, to form a parallelogram, we cut a piece of a rectangle from one side and placed it on the other side. Do you think we changed the area? The answer is no. All we did was rearrange it; the area of the new figure, the parallelogram, is the same as the original rectangle. So we have the Area of a parallelogram = bh. We have established that the area of a rectangle and a parallelogram to be A = bh. Be sure to spend some time identifying the width and the of a variety of parallelograms so students understand the difference. Area of Plane Figures Triangles Let s see how that helps us to understand the area formula for a triangle. If I were to cut the rectangle using its diagonal, I would have the following: h b Math 6 Notes Unit 8: Geometric Relationships Page 8 of 13 Revised 013 CCSS
Cutting the rectangle as shown above to create the triangle allows me to see that the triangle is half the rectangle. Therefore since the area of the rectangle is A = bh and the triangle is half of that rectangle, the 1 area of the triangle should be A = bh. We can easily see/count the rectangle contains 4 square units and the triangle contains 1 square units. Mathematically, we could show the area of the rectangle is A = 6 4 = 4units. The Area of the triangle = 1 bh or the Area of the triangle = bh. To compute the area we get 1 1 A = = = 6 4 4 1units or bh 6 4 4 A = = = = 1 square units. h h For this parallelogram, its is 4 units and its is 3 units. Therefore, the area is 4 3 = 1 units. If we draw a diagonal, it cuts the parallelogram into triangles. That means one triangle would have one-half of the area or 6 units. Note the and stay the same. So for a triangle, 1 1 ( )( ) A = bh, or 4 3 = 6 units Note: Trapezoids are NOT tested at the state level for 6 th grade. Area of a Plane Figure - Circles You can demonstrate the formula for finding the area of a circle. First, draw a circle; cut it out. Fold it in half; fold in half again. Fold in half two more times, creating 16 wedges when you unfold the circle. Cut along these folds. Math 6 Notes Unit 8: Geometric Relationships Page 9 of 13 Revised 013 CCSS
Rearrange the wedges, alternating the pieces tip up and down (as shown), to look somewhat like a parallelogram. radius (r) This is ½ of the distance around the circle or ½ of C. We know that : C = π r, so by substitution 1 1 C = π r 1 C = πr The more wedges we cut, the closer it would approach the shape of a parallelogram. No area has been lost (or gained). Our parallelogram has a of πr and a of r. We know from our previous discussion that the area of a parallelogram is bh. So we now have the area of a circle: A = bh A= (π r)( r) A= π r πr radius (r) Reflection - Check for Understanding For which plane figure(s) does the formula A = bh work? For which plane figure(s) does the formula A = lw work? 1 For which plane figure(s) does the formula A = bh work? For which plane figure(s) does the formula A= s work? For which plane figure(s) does the formula A= π r work? Perimeter of Plane Figures - Modeling Many of your students should know how to find the perimeter of most plane figures, but now is the time for a good review for them. Some teachers help students remember perimeter by capitalizing the RIM Math 6 Notes Unit 8: Geometric Relationships Page 10 of 13 Revised 013 CCSS
within and discussing that rim means the outer edge of an object or figure. A more formal definition of perimeter is the distance around the outside of a polygon. Conceptually, you might begin by showing the following figures and steps. Perimeter of rectangles P= + + + P= side l + side w + side l + side w P= + P= x length + x width or P = (length + width) Perimeter of Squares P = + + + P = side l + side w + side l + side w P= = 4 x side Perimeter of parallelograms P= + + + P= side l + side w + side l + side w P= + P= x length + x width or P = (length + width) Note in general, you can find the perimeter of any polygon by adding all the sides. P = sum of the sides. So, given a triangle for instance, like the one below: 6 cm P = 6 + 6 + 6 = or P = 6 3 P= 18 cm P = 18 cm Math 6 Notes Unit 8: Geometric Relationships Page 11 of 13 Revised 013 CCSS
7 mm P = 10 + 7 + 7 + 10 + 7 +7 = 48 mm or P = (10) + 4(7) = 0 + 8 = 48 mm 10mm Circumference of a circle Like perimeter, the circumference of a circle is the distance around the outside of a circle. To compute the circumference, students must know the diameter or the radius. The diameter is the line segment that passes through the center of a circle and has endpoints on the circle, or the length of that segment. The radius is the line segment with one endpoint at the center of the circle and the other endpoint on the circle, or the length of that segment. diameter radius Pi is the ratio of the circumference of a circle to the length of the diameter; π 3.14or. 7 There are many labs that will help students understand the concept of pi. Most involve measuring the distance around the outside of a variety of different size cans, measuring the diameter of the can and dividing to show that the relationship is 3 and something. You may want to use a table like the following to organize student work: Can # Circumference Diameter Circumference Diameter = π Once students record the measurements for a variety of cans, students should find the quotient of the circumference diameter is close to 3.14. Some quotients are not exact or are off due to measurement errors or lack of precision. So initially get students to comprehend the circumference diameter x 3. Math 6 Notes Unit 8: Geometric Relationships Page 1 of 13 Revised 013 CCSS
Two formulas should be introduced at this time: C = π diameter C = π radius and C = π d C = π r Discussions about the relationship of the diameter and radius, 1 diameter = radii and 1 radius = ½ diameter, help students realize the necessity for the two formulas. CCSS require students to KNOW formulas. Students need to begin memorizing them and knowing them fluently. Math 6 Notes Unit 8: Geometric Relationships Page 13 of 13 Revised 013 CCSS