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 10
This could be called ray XY and shown by XY. Y X Angles: Measuring and Classifying An angle can be seen as a rotation of a line about a fixed point. In other words, if I were to mark a point on a paper, then rotate a pencil around that point, I would be forming angles. One complete rotation measures 360º. Half a rotation would then measure 180º. A quarter rotation would measure 90º. Let s use a more formal definition. 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. Angles are measured in degrees ( ). Protractors are used to measure angles. Here are two interactive websites you might use to show students how to use this measuring tool. http://www.amblesideprimary.com/ambleweb/mentalmaths/protractor.html http://www.mathplayground.com/measuringangles.html 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. Math 6 Notes Unit 8: Geometric Relationships Page 2 of 10
acute right obtuse straight Angle Relationships Objectives: (6.12)The student will model the measures of complementary and supplementary angles with and without tools of measurement. (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. 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 2 1. What angle is vertical to 1? 2. Name an angle adjacent to 4. 3. If m 3 = 50, what are the measures of 1, 2, nd a 4? 4. What is the sum of the measures of 1, 2, 3, and 4? Math 6 Notes Unit 8: Geometric Relationships Page 3 of 10
1. 3is vertical with 1 2. Either 3 or 1 is correct 3. m 1 = 50 because vertical angles are congruent m 2 = 130 because 2 and 3 form a straight angle (180 50 ) m 4 = 130 because it is vertical with 2 and therefore congruent OR because 3 and 4 form a straight angle (180 50 ) 4. 360 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 f g f > l m g > Math 6 Notes Unit 8: Geometric Relationships Page 4 of 10
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 q 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 Objective: (6.9)The student will determine the measurement of missing angles of triangles based on the triangle sum theorem (sum of interior angles equals 180 ). The sum of the angle measures of any triangle is 180. This can be shown with a quick demonstration: 1) Draw and label a large triangle as shown. 2) Cut the triangle out. 3) Tear each angle from the triangle and place them so their vertices meet at a point. a a b a c b c b c Triangles can be classified by the measures of their angles: acute triangle 3 acute angles right triangle 1 right angle Math 6 Notes Unit 8: Geometric Relationships Page 5 of 10
obtuse triangle 1 obtuse angle Example: Classify each triangle by their angle measure: A 20 D G M 40 40 B 60 60 E 50 45 C F H J K 80 L ABC: ( ) m A= 180 60 + 60 m A= 180 120 m A= 60 Since all 3 angles are less than 90, ABC is an acute triangle. DEF: ( ) m E = 180 40 + 50 m E = 180 90 m E = 90 Since there is a right angle, is a right triangle. DEF GHJ: KLM: ( ) m H = 180 20 + 40 m H = 180 60 m MH = 180 120 ( 45 + 80 ) m M = 180 125 m M = 55 Since there is an obtuse angle, GHJ is an obtuse triangle. Since all 3 angles are acute, KLM is an acute triangle. 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 2 congruent sides scalene triangle no congruent sides equilateral isosceles scalene Example: Classify the triangle. The perimeter of the triangle is 15 cm. 10 cm 2.5 cm Math 6 Notes Unit 8: Geometric Relationships Page 6 of 10 x
Using the information given regarding the perimeter: x + 2.5 + 10 = 15 x + 12.5 = 15 x = 2.5 Since 2 sides are congruent, the triangle is isosceles. A tree diagram could also be used to show the triangle relationships. Tree Diagram for Triangles triangles acute obtuse right scalene isosceles scalene isosceles scalene isosceles equilateral Quadrilaterals Objectives: (6.5)The student will classify quadrilaterals. (6.6)The student will identify properties of quadrilaterals. (6.7)The student will compare regular and irregular quadrilaterals. (6.8)The student will draw regular and irregular quadrilaterals. A quadrilateral is a plane figure with four sides and four angles. They are classified based on congruent sides, parallel sides and right angles. Quadrilateral Type Definition Example >> Parallelogram Quadrilateral with both pairs of opposite sides parallel. >> Rhombus Parallelogram with four congruent sides. Note: This polygon is a parallelogram. Math 6 Notes Unit 8: Geometric Relationships Page 7 of 10
Rectangle Parallelogram with four right angles. Note: This polygon is a parallelogram. Square Parallelogram with four right angles and four congruent sides. Note: This polygon is a parallelogram. Trapezoid Quadrilateral with exactly one pair of parallel sides. >> >> Another way to show the relationship of the parallelograms is to complete a Venn diagram as shown below. parallelograms rectangles squares rhombi Vocabulary becomes very important when trying to solve word problems about quadrilaterals. Example: A quadrilateral has both pairs of opposite sides parallel. One set of opposite angles are congruent and acute. The other set of angles is congruent and obtuse. All four sides are NOT congruent. Which name below best classifies this figure? A. parallelogram B. rectangle C. rhombus D. trapezoid We have both pairs of opposite sides parallel, so it cannot be the trapezoid. Since the angles are not 90 in measure, we can rule out the rectangle. We are told that the 4 sides are not congruent, so it cannot be the rhombus. Therefore, we have a parallelogram. Math 6 Notes Unit 8: Geometric Relationships Page 8 of 10
The word regular has a very powerful meaning in geometry. A regular quadrilateral is one in which all the sides are of equal length and all the angles have the same measure. That limits us to one type of quadrilateral the square! When the student investigates polygons, regular polygons will be discussed. CRT Example: CRT Example: Polygons Objectives: (6.5)The student will classify polygons. (Accelerated Only) (6.6)The student will identify properties of polygons. (Accelerated Only) (6.7)The student will compare regular and irregular polygons. (Accelerated Only) (6.8)The student will draw regular and irregular polygons. (Accelerated Only) A polygon is defined as a closed geometric figure formed by connecting line segments endpoint to endpoint. Polygons Not Polygons Math 6 Notes Unit 8: Geometric Relationships Page 9 of 10
A regular polygon is one in which all the sides are of equal length and all the angles have the same measure. Examples: Regular pentagon with all sides and all angles congruent B A C Not a regular polygon as only 2 sides are congruent. Note: When labeling geometric figures, mark angles and segments that are equal in measure with similar marks. For example, in the pentagon all the angles are marked equal with an arc with one slash and the sides marked equal with one slash. In triangle ABC, the measure of AB and AC are shown equal with two slashes. Polygons are named by the number of sides. We know a triangle has 3 sides. Below are the names of other polygons. Polygons Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon # of sides 4 5 6 7 8 9 10 Have you ever had a problem with drawing a polygon? Many times we end up with everything bunched at the end, or not a polygon. Here is a suggestion to more easily draw a polygon: 1. Lightly draw a circle. 2. Place points on the circle to represent the endpoints of the segments of the polygon you wish to create. 3. Connect the points to create your polygon. 4. Erase the circle. You have your polygon! Math 6 Notes Unit 8: Geometric Relationships Page 10 of 10