Math 7, Unit 8: Geometric Figures Notes

Math 7, Unit 8: Geometric Figures Notes Points, Lines and Planes; Line Segments and Rays s 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. 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. line extends indefinitely. line, containing infinitely many points, is considered to be a set of points, hence it has no thickness. 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. plane is a flat surface. Such things as table tops, desks, windowpanes, and walls suggest planes. plane, like the aforementioned, does not have thickness and extends indefinitely. C B plane is named by 3 points that are on the plane (but not on the same line called noncollinear points). This plane could be called plane BC, or plane CB or plane CB. line segment contains two endpoints and all the points between those endpoints. B line segment is named by its endpoints. The above example could be read line segment B or B, which is also read line segment B. 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 7 Notes Unit 8: Geometric Figures Page 1 of 16

This could be called ray XY and shown by XY. Y X We will need to have a system for identifying line segments that have the same length. Line segments are congruent if they have the same length. Two figures are congruent if they have the same shape and size. We use the symbol to mean is congruent to. We will use tick marks to indicate congruent line segments. In the rectangle shown, line segment WX and line segment YZ are congruent; line segment WZ is congruent to line segment XY. WX YZ WZ XY W X Z Y ngles: Measuring and Classifying n 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. quarter rotation would measure 90º. Let s use a more formal definition. n angle is formed by the union of two rays with a common endpoint, called the vertex. ngles 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. Math 7 Notes Unit 8: Geometric Figures Page 2 of 16

ngles 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. cute 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. nd finally, straight angles measure 180º. acute right obtuse straight We call two angles whose sum is 90º complementary angles. For instance, if m P = 40 and m Q = 50, then P and Q are complementary angles. If m = 30, then the complement of 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: 3 2 4 5 1 1. Name an angle that is complementary to 1. 2. Name an angle that is supplementary to 1. 3. If m 2 = 50, what is the measure of 1? 4. What is the measure of 4? 5. What is the sum of the measures of 1, 2, 3, 4 nd a 5?? 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 7 Notes Unit 8: Geometric Figures Page 3 of 16

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. ngle Relationships Objective: (6.12)The student will describe properties of parallel and perpendicular lines. (6.15)The student will verify congruent angles and parallel and perpendicular lines. djacent 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. 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 7 Notes Unit 8: Geometric Figures Page 4 of 16

ngles will be shown as congruent by using tick marks again. If angles are marked with the same number of tick marks, then the angles are congruent. P R Q TQP RQS TQS PQR T S transversal is a line that intersects two or more lines. In the figure to the right, line t is a transversal. If I asked you to look at the figure and find two angles that are on the same side of the transversal, one an interior angle (between the lines), the other an exterior angle, that were not adjacent, could you do it? 2 and 3are on the same side of the transversal, one interior, the other is exterior, but they are adjacent. These angles do not fit the conditions. 1 2 3 4 5 6 8 7 t m n How about 2 and 6? These two angles do fit the conditions. We call these angles corresponding angles. Can you name any other pairs of corresponding angles? If you said 3 and 7, 4 and 8, 1 and 5, you d be right. When a pair of parallel lines is cut by a transversal, there is a special relationship between the corresponding angles. Discover what this relationship is! 1 2 4 3 5 6 8 7 1) Use the lines on a piece of graph paper or a piece of lined paper as a guide to draw a pair of parallel lines. You can also use both edges of your straightedge to create parallel lines on a blank sheet of paper. 2) Draw a transversal intersecting the parallel lines. Label the angles with numbers as in the diagram above. 3) Measure 1. Calculate the measures of the other three angles that share the same vertex. 4) Measure 8 and calculate the measures of the other three angles. Record your findings into the table. ngle Measure ngle Measure 1 5 2 6 3 7 4 8 Math 7 Notes Unit 8: Geometric Figures Page 5 of 16

Look for patterns. Complete the conjectures below: If two parallel lines are cut by a transversal, then the corresponding angles are. Hopefully, you discovered that the corresponding angles created when a transversal intersects parallel lines are congruent! This will save us time when finding angle measures. Example: m 2 = 50. Find the measure of all other angles. m 1 = 130, since 1 and 2 are adjacent and supplementary m 3 = 130, since 1 and 3are vertical angles or 2 and 3 are adjacent and supplementary a 1 2 3 4 b 5 6 8 7 a b m 4 = 50, since 2 and 4 are vertical angles or 1 and 4 are adjacent and supplementary m 5 = 130, since 1 and 5 are corresponding angles m 7 = 130, m 6 = 50, and m 8 = 50 (using same reasoning as above) Circles circle is defined as all points in a plane that are equal distance (called the radius, C) from a fixed point (called the center of the circle, point C). We name circles by its center point. For example, if the center of the circle is point C, then the circle is called circle C. The distance across the circle, through the center, is called the diameter( DB ). D E C F B chord in a circle is a line segment whose endpoints are on the circle ( EF ). n arc is a continuous part of the circle that is named by its endpoints( EF ). central angle is an angle formed by 2 radii ( DC). sector of a circle is that part of the circle enclosed by an arc ( D ) and 2 radii ( DC and C ). sector is the shaded region shown in the circle to the right. D C Math 7 Notes Unit 8: Geometric Figures Page 6 of 16

Example: Determine which of the following are true or false for the circle given. 1. FG is a diameter of the circle. false F 2. GJ is a radius of the circle. true 3. HK and HJ are both chords of the circle. true H G 4. HGF and FGJ are supplementary angles. true Fill in the blank: K 5. The central angle that is part of the sector shaded in the circle is named. FGJ 6. If m HGF = 120, the measure of FGJ is. 60 7. If the shaded sector represents 20% of the circle graph, what would be the measure of the central angle? 20% of 360 = 72 J Classifying Polygons Objectives: (6.7)The student will draw regular and irregular polygons. (6.8)The student will identify regular and irregular polygons. (6.9)The student will describe regular and irregular polygons. (6.10)The student will classify regular and irregular polygons by a variety of attributes. (6.11)The student will compare regular and irregular polygons. polygon is defined as a closed geometric figure formed by connecting line segments endpoint to endpoint. Polygons Not Polygons regular polygon is one in which all the sides are of equal length and all the angles have the same measure. Note: When labeling geometric figures, mark angles and Examples: 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 BC, the measure of B and C are B C shown equal with two slashes. Regular pentagon with Not a regular polygon all sides and all angles as only 2 sides are congruent congruent. Math 7 Notes Unit 8: Geometric Figures Page 7 of 16

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! Classifying 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 Math 7 Notes Unit 8: Geometric Figures Page 8 of 16

Example: Classify each triangle by their angle measure: 20 D G 60 40 40 60 60 E 50 120 B 45 C F H J K M 55 80 L 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. 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. x 10 cm 2.5 cm 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 Math 7 Notes Unit 8: Geometric Figures Page 9 of 16

Classifying Quadrilaterals 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. 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. >> >> nother way to show the relationship of the parallelograms is to complete a Venn diagram as shown below. parallelograms rectangles squares rhombi Math 7 Notes Unit 8: Geometric Figures Page 10 of 16

Vocabulary becomes very important when trying to solve word problems about quadrilaterals. Example: 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. ll four sides are NOT congruent. Which name below best classifies this figure?. 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. ngle Measures in Polygons Objective: (6.21)The student will determine the sum of the interior angles of polygons. 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 The sum of the angle measures in any quadrilateral can be found by dividing the figure into 2 triangles. Since the sum of the angle measures in each of these triangles is 180, we can reason that the sum of the angle measures in our quadrilateral would be 180 + 180 = 360 We can continue this reasoning with other polygons. Math 7 Notes Unit 8: Geometric Figures Page 11 of 16

Finish identifying the polygons below by name; then complete drawing all the diagonals from one vertex of the polygon. Triangle Quadrilateral Number of sides of polygon 3 4 5 6 7 8 9 Triangles formed by diagonals from one vertex 1 2 Sum of measures of angles 1 180 = 180 2 180 = 360 Example: Find the unknown angle measure in the quadrilateral. 52 We know the sum of the angle measures in a quadrilateral is 360. 137 x + 90 + 52 + 137 = 360 x + 279 = 360 x = 81 x The missing angle is 81. Example: Find the sum of the angle measures in a decagon. Divide the polygon into triangles to show how that reasoning could get you to the answer. decagon has 10 sides. From one vertex we can diagonals to create 8 triangles. 8 180 = 1440 There are 1440 in a decagon. 3 2 1 4 5 6 7 8 Math 7 Notes Unit 8: Geometric Figures Page 12 of 16

Identifying Congruent Figures We know two figures are congruent if they have the same size and shape. The tiles on a floor are usually congruent to each other; the desk tops in your classroom are probably congruent to each other. If all the corresponding sides and angles of two polygons are congruent, then the polygons are congruent. There is a short cut we can use with triangles only. If all the corresponding sides of a triangle are congruent, then the triangles are congruent. This is called side-side-side ; a geometry course will go into this in much more detail. For us, it is a way to determine whether or not triangles are congruent by just looking at the sides (we do not have to check the angle measures). Please note: this works for triangles ONLY. Example: re the 2 triangles congruent? O Yes, since all 3 corresponding sides are congruent. We write CT DOG. C T G D Example: re the quadrilaterals congruent? Maybe we cannot be sure until we know if all the corresponding angles are congruent. 2 m 4 m 4 m 2 m 4 m 4 m 2 m In the second group, we can now see that all of the corresponding angles are congruent; therefore, the quadrilaterals are congruent. 2 m 4 m 4 m 135 45 45 135 4 m 2 m 4 m 2 m One of the values of knowing that two figures are congruent is the ability to find missing measures in figures. Example: Determine the missing measures in each set of congruent polygons. Since the figures are congruent, the corresponding angles and sides are congruent. Therefore, x = 110 and y = 5 inches. 5 in. 70 5 in. 5 in. 70 5 in. 110 110 x 110 y 70 5 in. 5 in. 70 5 in. Math 7 Notes Unit 8: Geometric Figures Page 13 of 16

Transformations Objectives: (6.18)The student will describe the location of the original figure and its transformation on a coordinate plane. (6.22)The student will explore transformations using coordinate geometry. Geometry not only looks at figures, it also studies the movement of figures. If you move all the points of one figure to create a new geometric figure, you call the new figure the image. Each point of the image matches exactly with a corresponding point of the original figure; in other words, the image is congruent to the original figure. This changing of position is called a transformation. There are 3 types of transformations we are going to study: translations, rotation, and reflection. You may be more familiar with these as being called slide, turn, and flip. translation is the simplest. If you copy a figure onto a piece of paper, than slide the paper along a straight path without turning it, your slide motion represents a translation. rotation is a turning motion. Points of the original figure rotate an identical number of degrees around a fixed center pint. reflection flips the figure across a line of reflection, creating a mirror image. Math 7 Notes Unit 8: Geometric Figures Page 14 of 16

Transformations can also be shown on a coordinate grid. translation will move each point like an ordered pair. For example, the translation shown moved the triangle right 3 units and up 4 units, which we could also show x+ 3, y+ 4. as ( ) ' C' B' prime mark (') is used with the label of an image point. The image of point is shown as point ' (read prime ). C B reflection can be shown on a coordinate grid by reflecting the figure across an axis. B G E F cross the x-axis, the figure will follow the rule x, y. ( ) C C' D D' ' B' E' F' G' cross the x-axis, the figure will follow the rule xy,. ( ) Math 7 Notes Unit 8: Geometric Figures Page 15 of 16

rotation can be shown on a coordinate grid by moving the figure about a point. Unless it is indicated otherwise, rotation occurs around the origin. This example shows a rotation counterclockwise about the origin. F' G' ' E' C' D' B' F G E B C D rotation can also occur around a given point. This example rotated the triangle 180 about point X. Z Y X Y' Z' Math 7 Notes Unit 8: Geometric Figures Page 16 of 16