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Geometry Points, Lines, Planes, & Angles Table of Contents Points, Lines, & Planes Line Segments Pythagorean Theorem Distance between points Area of Figures in Coordinate Plane Midpoint formula Locus & Constructions Angles & Angle Addition Postulate Angle Pair Relationships Angle Bisectors & Constructions Table of Contents for Videos Demonstrating Constructions Congruent Segments Midpoints Circle Equilateral Triangles Congruent Angles Angle Bisectors Points, Lines, & Planes Definitions An "undefined term" is a word or term that does not require further explanation. There are three undefined terms in geometry: Points - A point has no dimensions (length, width, height), it is usually represent by a capital letter and a dot on a page. It shows position only. Lines - composed of an unlimited number of points along a straight path. A line has no width or height and extends infinitely in opposite directions. Planes - a flat surface that extends indefinitely in two- dimensions. A plane has no thickness. Points & Lines A television picture is composed of many dots placed closely together. But if you look very closely, you will see the spaces. However, in geometry, a line is composed of an unlimited (infinite) number of points. There are no spaces between the points that make a line. .................................... A B You can always find a point between any two other The line above would be called line or line Points & Lines Line Points are labeled with letters. , , or all refer to the same line Line a Lines are named by using any two points OR by using a single lowercase letter. Arrowheads show that the line continues without end in opposite directions. Collinear Points Line , , or all refer to the same line Points A, B, and C are NOT collinear points since they do not lie on the same (one) line. Line a Points D, E, and F are called collinear points, meaning they all lie on the same line. Postulate: Any two points are always collinear. Answer Example: Give six different names for the line that contains points U, V, and W. Intersecting Lines If two non-parallel lines intersect in a plane they do so at only one point. and intersect at K. Postulate: two lines intersect at exactly one point. Example Answer a. Name three points that are collinear b. Name three sets of points that are noncollinear c. What is the intersection of the two lines? Rays or Rays are also portions of a line. Rays start at an initial point, here endpoint A, and continues infinitely in one direction. is read "ray AB" Naming Rays or Rays and are NOT the same. They have different initial points and extend in different directions. Opposite Rays Suppose point C is between points A and B Rays and are opposite rays. Opposite rays are two rays with a common endpoint that point in opposite directions and form a straight line. Collinear Rays Recall: Since A, B, and C all lie on the same line, we know they are collinear points. Similarly, segments and rays are called collinear, if they lie on the same line. Segments, rays, and lines are also called coplanar if they all lie on the same plane. Example Name a point that is collinear with the given points. a. R and P b. M and Q c. S and N d. O and P Example Name two opposite rays on the given line e. f. g. h. 1 is the same as . True Answer False Hint Read the notation carefully. Are they asking about lines, line segments, or rays? click 2 is the same as . True Answer False 3 Line p contains just three points. True Answer False Hint click to reveal Remember that even though only three points are marked, a line is composed of an infinite number of points. You can always find another point in between two other points. Points D, H, and E are collinear. True False Answer 4 5 Points G, D, and H are collinear. True Answer False 6 Ray LJ and ray JL are opposite rays. Explain your answer. Yes Answer No Which of the following are opposite rays? A B and and C and D and Answer 7 A J B K C L Answer Name the initial point of 8 A J B K C L Answer Name the initial point of 9 Example Are the three points collinear? If they are, name the line they lie on. a. L, K, J b. N, I, M c. M, N, K d. P, M, I Planes Planes can be named by any three noncollinear points: Plane KMN, plane LKM, or plane KNL or, by a single letter such as Plane R. (These all name the same plane) Coplanar points are points that lie on the same plane: Points K, M, and L are coplanar. Points O, K, and L are non-coplanar in the diagram above. However, you could draw a plane to contain any three points Collinear points are points that are on the same line. J,G, and K are three collinear points. F,G, and H are three collinear points. J,G, and H are three non-collinear points. Any three non-collinear points can name a plane. Coplanar points are points that lie on the same plane. F, G, H, and I are coplanar. F, G, H, and J are also coplanar, but the plane is not drawn. F,G, and H are coplanar in addition to being collinear. G, I, and K are non-coplanar and non-collinear. Intersecting Planes The intersection of these two planes is shown by line A B If two planes intersect, they intersect along exactly one line. As another example, picture the intersections of the four walls in a room with the ceiling or the floor. You can imagine a line laying along the intersections of these planes. Through any three non-collinear points there is exactly one plane. Example Name the following points: A point not in plane HIE A point not in plane GIE Two points in both planes Two points not on 10 Line BC does not contain point R. Are points R, B, and C collinear? No Answer Yes 11 Plane LMN does not contain point P. Are points P, M, and N coplanar? Yes Answer No Hint: click to reveal What do we know about any three points? 12 Plane QRS contains a picture) . Are points Q, R, S, and V coplanar? (Draw Yes Answer No 13 Plane JKL does not contain . Are points J, K, L, and N coplanar? Yes Answer No and intersect at A Point A B Point B C Point C D Point D Answer 14 Which group of points are noncoplanar with points A, B, and F on the cube below. A E, F, B, A B A, C, G, E C D, H, G, C D F, E, G, H Answer 15 16 Are lines and coplanar on the cube below? Yes Answer No Plane ABC and plane DCG intersect at _____? A C B line DC C Line CG D they don't intersect Answer 17 Planes ABC, GCD, and EGC intersect at _____? A line B point C C point A D line Answer 18 19 A B B C C D D F Answer Name another point that is in the same plane as points E, G, and H 20 A H B B C D D A Answer Name a point that is coplanar with points E, F, and C A Always B Sometimes C Never Answer Intersecting lines are __________ coplanar. 21 Two planes ____________ intersect at exactly one point. A Always B Sometimes C Never Answer 22 A plane can __________ be drawn so that any three points are coplaner A Always B Sometimes C Never Answer 23 A plane containing two points of a line __________ contains the entire line. A Always B Sometimes C Never Answer 24 Four points are ____________ noncoplanar. A Always B Sometimes C Never Answer 25 Two lines ________________ meet at more than one point. A Always B Sometimes C Never Look what happens if I place line y directly on top of line x. click to reveal HINT Answer 26 Line Segments Line Segments or Line segments are portions of a line. or endpoint endpoint is read "segment AB" Line Segment or are different names for the same segment. It consists of the endpoints A and B and all the points on the line between them. Ruler Postulate On a number line, every point can be paired with a number and every number can be paired with a point. Coordinates indicate the point's position on the number line. The symbol AF stands for the length of . This distance from A to F can be found by subtracting the two coordinates and taking the absolute value. A -10 -9 -8 B -7 -6 C -5 -4 D -3 -2 -1 E 0 1 F 2 3 4 5 6 7 A F coordinate coordinate Distance AF = |-8 - 6| = 14 8 9 10 Why did we take the Absolute Value when calculating distance? In our previous slide, we were seeking the distance between two points. Distance is a physical quantity that can be measured - distances cannot be negative. A -10 -9 -8 B -7 -6 C -5 -4 D -3 -2 -1 E 0 1 F 2 3 4 5 6 7 A F coordinate coordinate 8 9 10 Distance AF = |-8 - 6| = 14 When you take the absolute value between two numbers, the order in which you subtract the two numbers does not matter. Definition: Congruence Equal in size and shape. Two objects are congruent if they have the same dimensions and shape. Roughly, 'congruent' means 'equal', but it has a precise meaning that you should understand completely when you consider complex shapes. Congruent Segments Line Segments are congruent if they have the same length. Congruent lines can be at any angle or orientation on the plane; they do not need to be parallel. Read as: "The line segment DE is congruent to line segment HI." Constructing Congruent Segments Given: Given: AB 1. Draw a reference line w/ your straight edge. Place a reference point to indicate where your new segment start on the line. 2. Place your compass point on point A. 3. Stretch the compass out so that the pencil tip is on point B. Constructing Congruent Segments (cont'd) 4. Without stretching out your compass, place the compass point on the reference point & swing the pencil so that it crosses the reference line. Constructing Congruent Segments (cont'd) 5. Make a point where the pencil crossed your line. Label your new segment. Constructing Congruent Segments Try This! Construct a Congruent Segment on the horizontal line. Teacher Notes 1) Constructing Congruent Segments Try This! Construct a Congruent Segment on the slanted line. 2) Video Demonstrating Constructing Congruent Segments using Dynamic Geometric Software Click here to see video Definition: Parallel Lines Lines are parallel if they lie in the same plane, and are the same distance apart over their entire length. That is, they do not intersect. Example Find the measure of each segment in centimeters. cm a. = 8click- 2 = 6 cm b. = 1.5 cm click 27 Find a segment that is 4 cm long B C D cm Answer A 28 Find a segment that is 6.5 cm long A C D cm Answer B 29 Find a segment that is 3.5 cm long B C D cm Answer A 30 Find a segment that is 2 cm long B C D cm Answer A 31 Find a segment that is 5.5 cm long B C D cm Answer A If point F was placed at 3.5 cm on the ruler, how from point E would it be? A 5 cm B 4 cm C 3.5 cm D 4.5 cm cm Answer 32 Segment Addition Postulate If B is between A and C, Then AB + BC = AC. Or, If AB + BC = AC, then B is between A and C. AB BC AC Simply said, if you take one part of a segment (AB), and add it to another part of the segment (BC), you get the entire segment. The whole is equal to the sum of its parts. Example The segment addition postulate works for three or more segments if all the segments lie on the same line (i.e. all the points are collinear). In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6 Find CD and BE Start by filling in the information you are given AE AB BC CD DE In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6 27 || x Can you finish the rest? 6 || x 5 2x+11 = 27 2x = 16 click x = 8 = CD click click =click19 Example K, M, and P are collinear with P between K and M. PM = 2x + 4, MK = 14x - 56, and PK = x + 17. Solve for x 2) From the segment addition postulate, we know that KP + PM = MK (the parts equal the whole) 3) Solve for x (x + 17) + (2x + 4) = 14x - 56 Answer 1) Draw a diagram and insert the information given into the diagram Example 1) First, arrange the points in order and draw a diagram a) BPM b) BPLM 2) Segment addition postulate gives 3x+13 + 2x+11 + 3x+16 = 3x+140 3) Combine like terms and isolate/solve for the variable x Answer P, B, L, and M are collinear and are in the following order: a) P is between B and M b) L is between M and P Draw a diagram and solve for x, given: ML = 3x +16, PL = 2x +11, BM = 3x +140, and PB = 3x + 13 For next six questions we are given the following information about the collinear points: Hint: always start these problems by placing the information you have into the diagram. We are given the following information about the collinear points: What is , , and ? Answer 33 We are given the following information about the collinear points: What is ? Answer 34 We are given the following information about the collinear points: What is ? Answer 35 We are given the following information about the collinear points: What is ? Answer 36 We are given the following information about the collinear points: What is ? Answer 37 We are given the following information about the collinear points: What is ? Answer 38 X, B, and Y are collinear points, with Y between B and X. Draw a diagram and solve for x, given: BX = 6x + 151 XY = 15x - 7 BY = x - 12 Answer 39 Q, X, and R are collinear points, with X between R and Q. Draw a diagram and solve for x, given: XQ = 15x + 10 RQ = 2x + 131 XR = 7x +1 Answer 40 B, K, and V are collinear points, with K between V and B. Draw a diagram and solve for x, given: KB = 5x BV = 15x + 125 KV = 4x +149 Answer 41 The Pythagorean Theorem Pythagorean Theorem c b c2 = a2 + b2 a Pythagoras was a philosopher, theologian, scientist and mathematician born on the island of Samos in ancient Greece and lived from c. 570–c. 495 BC. The theorem that in a right triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides. Pythagorean Theory Visual Proof 1 Pythagorean Theory Visual Proof 2 Using the Pythagorean Theorem In the Pythagorean Theorem, c always stands for the longest side. In a right triangle, the longest side is called the hypotenuse. The hypotenuse is the side opposite the right angle. 25 = a2 + 9 5 c2= a2 + b2 -9 a=? -9 16 = a2 = a 4 = 3 You will use the Pythagorean Theorem often. a Answer Example What is the length of side c? Answer 42 The longest side of a triangle is called the ___________ What is the length of side a? Answer 43 Hint: click to reveal Always determine which side is the hypotenuse first What is the length of c? B Answer 44 What is the length of the missing side? Answer 45 What is the length of side b? Answer 46 What is the measure of x? x 8 17 Answer 47 Pythagorean Triples are three positive integers for side lengths that satisfy a2 + b2 = c2 ( 3 , 4 , 5 ) ( 5, 12, 13) (6, 8, 10) ( 7, 24, 25) ( 8, 15, 17) ( 9, 40, 41) (10, 24, 26) (11, 60, 61) (12, 35, 37) (13, 84, 85) etc. There are many more. Remembering some of these combinations may save you some time 48 A triangle has sides 30, 40 , and 50, is it a right triangle? Yes Answer No 49 A triangle has sides 9, 12 , and 15, is it a right triangle? Yes Answer No A triangle has sides √3, 2 , and √5, is it a right triangle? 50 Yes Answer No Distance Between Points Computing distance Computing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles. Computing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle. The Pythagorean Theorem is true for all right triangles. If we know the lengths of two sides of a right triangle then we know the length of the third side. Relationship between the Pythagorean Theorem & Distance Formula The Pythagorean Theorem states a relationship among the sides of a right triangle. The distance formula calculates the distance using the points' coordinates. (x2, y2) c2= a2 + b2 c c a b (x1, y1) (x2, y1) The Distance Formula The distance 'd' between any two points with coordinates is given (x2, y2) by the formula: (x1, y1)and d= The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula. Note: recall that all coordinates are (x-coordinate, y-coordinate). Example Calculate the distance from Point K to Point I (x1, y1) (x2, y2) d= Plug the coordinates into the distance formula KI = Answer Label the points - it does not matter which one you label point 1 and point 2. Your answer will be the same. 51 Calculate the distance from Point J to Point K A B C Answer D 52 Calculate the distance from H to K A B D Answer C Calculate the distance from Point G to Point K 53 A B C Answer D 54 Calculate the distance from Point I to Point H A B C Answer D 55 Calculate the distance from Point G to Point H A B C Answer D Area of Figures in the Coordinate Plane Calculating Area of Figures in the Coordinate Plane Area Formulas: Area of a Triangle: A = bh 1 2 Area of a Rectangle: A = lw Steps to calculate the area: 1) Calculate the desired distances using the distance formula > Ex: base & height in a triangle > Ex: length & width in a rectangle 2) Calculate the area of the shape Example: Calculate the area of the rectangle. Steps to calculate the area: 1) Calculate the desired distances using the distance formula > = FG & w = EF = 2) Calculate the area of the shape =w Steps to calculate the area: 1) Calculate the desired distances using the distance formula > b = AC & h = BD Answer Example: Calculate the area of the triangle. Answer Example: Calculate the area of the triangle. 2) Calculate the area of the shape 56 Calculate the area of the rectangle. A C 10 D 20 Answer B Calculate the area of the triangle. A 75 B 144 C 84.5 D 169 Answer 57 Calculate the area of the rectangle. D A 10,000 B 100 C 50 Answer 58 59 Calculate the area of the triangle. A C 22.5 D 45 Answer B Midpoint Formula Midpoint of a line segment A number line can help you find the midpoint of a segment. The midpoint of GH, marked by point M, is -1. Here's how you calculate it using the endpoint coordinates. Take the coordinates of the endpoint G and H, add them together, and divide by two. = = -1 Midpoint Formula Theorem The midpoint of a segment joining points with coordinates the point (x2,with y2) coordinates and (x1, is y1) Calculating Midpoints in a Cartesian Plane Segment PQ contain the points (2, 4) and (10, 6). The midpoint M of is the point halfway between P and Q. Just as before, we find the average of the coordinates. Remember that points are written with the xcoordinate first. (x, y) The coordinates of M, the midpoint of PQ, are (6, 5) ( , ) Find the midpoint coordinates (x,y) of the segment and B(5,6) A (4, 3) B (3, 4) C (6, 8) D (2.5, 3) Hint: Always label the points coordinates first click connecting points A(1,2) Answer 60 61 A (-1, -1) B (-3, -8) C (-8, -3) D (1, 1) Answer Find the midpoint coordinates (x,y) of the segment connecting the points A(2,5) and B(4, -3) 62 A (-1, 1) B (1, 1) C (-1, -1) D (1, -1) Answer Find the coordinates of the midpoint (x, y) of the segment with endpoints R(-4, 6) and Q(2, -8) 63 A (-3, 3) B (6, -4) C (-4, 6) D (4, 6) Answer Find the coordinates (x, y) of the midpoint of the segment with endpoints B(-1, 3) and C(-7, 9) Find the midpoint (x, y) of the line segment between A(-1, 3) and B(2,2) A (3/2, 5/2) B (1/2, 5/2) C (1/2, 3) D (3, 1/2) Answer 64 Example: Finding the coordinates of an endpoint of an segment Use the midpoint formula to write equations using x and y. Find the other endpoint of the segment with the endpoint (7,2) and midpoint (3,0) A (-1, -2) B (-2, -1) C (4, 2) D (2, 4) Answer 65 Find the other endpoint of the segment with the endpoint (1, 4) and midpoint (5, -2) A (11, -8) B (9, 0) C (9, -8) D (3, 1) Answer 66 Locus & Constructions Introduction to Locus Definition: Locus is a set of points that all satisfy a certain condition, in this course, it is the set of points that are the same distance from something else. Theorem: locus between two points All points on the perpendicular bisector of a line segment connecting two points are equidistant from the two points. Theorem: locus from a given line All points equidistant from a given line is a parallel line. Theorem: locus between two lines The locus of points equidistant from two given parallel lines is a parallel line midway between them. locus: equidistant from two points The locus of points equidistant from two points, A and B, is the perpendicular bisector of the line segment determined by the two points. X Y The distance (d) from point A to the locus is equal to the distance (d') from Point B to the locus. The set of all these points forms the red line and is named the locus. Point M is the midpoint of . X is equidistant (d=d') from A and B. Y lies on the locus; it is also equidistant from A and B. Bisect a Line Segment We can find the midpoint of a line segment by constructing the locus between two points. Given Bisect a Line Segment We can find the midpoint of a line segment by constructing the locus between two points. Given 1. Place the point of the compass on A and open it so the pencil is more than halfway, it does not matter how far, to point B Bisect a Line Segment We can find the midpoint of a line segment by constructing the locus between two points. Given 2. Keeping the compass locked, place the point of the compass on B and draw an arc so that it intersects the one you just drew Bisect a Line Segment We can find the midpoint of a line segment by constructing the locus between two points. Given M 3. Draw a line through the arc intersections. M marks the midpoint of Bisect a Line Segment We can find the midpoint of a line segment by constructing the locus between two points. 2. Keeping the compass locked, place the point of the compass on B and draw an arc so that it intersects the one you just drew Given M 1. Place the point of the compass on A and open it so the pencil is more than halfway, it does not matter how far, to point B 3. Draw a line through the arc intersections. M marks the midpoint of Bisect a Line Segment Try this! Construct the midpoint of the segment below. 3) Bisect a Line Segment Try this! Construct the midpoint of the segment below. 4) Bisect a Line Segment w/ a string & rod We can also construct the locus between two points with string, a rod (e.g. marker, closed pen, etc) & a pencil. Given 1. Place the rod on A and extend your piece of string out. The pencil should be more than halfway across the segment. It does not matter how far, to point B. Bisect a Line Segment w/ a string & a rod We can also construct the locus between two points with string, a rod (e.g. marker, closed pen, etc) & a pencil. Given 2. Place the rod on B and extend your piece of string out. The pencil should be more than half way across the segment. It does not matter how far, to point A. Bisect a Line Segment w/ a string & a rod We can also construct the locus between two points with string, a rod (e.g. marker, closed pen, etc) & a pencil. Given M 3. Draw a line through the arc intersections. M marks the midpoint of Bisect a Line Segment w/ a string & a rod We can find the midpoint of a line segment by constructing the locus between two points. 2. Place the rod on B and extend your piece of string out. The pencil should be more than half way across the segment. It does not matter how far, to point A. Given M 1. Place the rod on A and extend your piece of string out. The pencil should be more than halfway across the segment. It does not matter how far, to point B. 3. Draw a line through the arc intersections. M marks the midpoint of Bisect a Line Segment w/ a string & a rod Try this! Construct the midpoint of the segment below. 5) Bisect a Line Segment w/ a string & a rod Try this! Construct the midpoint of the segment below. Bisect a Line Segment w/ Paper Folding 3. Unfold your patty paper. Draw a line along the fold. Other paper edge 2. Fold your patty paper so that point D lines up with point E. Crease the fold. FOLD 1. On patty paper, draw a segment. Label the endpoints D & E. 4. Draw & label your midpoint M. Bisect a Line Segment w/ Paper Folding Try this! Construct the midpoint of the segment below. 7) Bisect a Line Segment w/ Paper Folding Try this! Construct the midpoint of the segment below. 8) Constructions Dividing a line segment into x congruent segments. Let us divide AB into 3 equal segments - we could choose any number of segments. 1. From point A, draw a line segment at an angle to the given line, and about the same length. The exact length is not important. 2. Set the compass on A, and set its width to a bit less than 1/3 of the length of the new line. Step the compass along the line, marking off 3 arcs. Label the last C. 3. Set the compass width to CB. 4. Using the compass set to CB, draw an arc below A 5. With the compass width set to AC, draw an arc from B intersecting the arc you just drew in step 4. Label this D. C D 6. Draw a line connecting B with D 7. Set the compass width back to AC and step along DB making 3 new arcs across the line 8. Draw lines connecting the arc along AC and BD. These lines intersect AB and divide it into 3 congruent segments. Example: Construction Divide the line segment into 3 congruent segments. 67 Point C is on the locus between point A and point B True Answer False 68 Point C is on the locus between point A and point B True Answer False How many points are equidistant from the endpoints of ? A 2 B 1 C 0 D infinite Answer 69 You can find the midpoint of a line segment by A measuring with a ruler B constructing the midpoint C finding the intersection of the locus and line segment D all of the above Answer 70 The definition of locus A a straight line between two points B the midpoint of a segment C the set of all points equidistant from two other points D a set of points Answer 71 Videos Demonstrating Constructing Midpoints using Dynamic Geometric Software Click here to see video using compass and segment tool Click here to see video using menu options Construction of a Circle Answer What is a circle? Extension: Construction of a Circle 1. Find the midpoint of a line segment 1st, using any of the methods that we discussed. Teacher Notes M Extension: Construction of a Circle 2. Take your compass (or rod, string & pencil) and place the tip (or rod) on your Midpoint. Extend out your pencil as much as you would like. Turn your compass (or string & pencil) 3600 to make your circle. M Construction of a Circle Try this! Create a circle using the segment below. 9) Construction of a Circle Try this! Create a circle using the segment below. 10) Video Demonstrating Constructing a Circle using Dynamic Geometric Software Click here to see video Construction of an Equilateral Triangle Answer What is an equilateral triangle? Extension: Construction of an Equilateral Triangle 1. Construct a segment of any length. 2. Take your compass and place the tip on one endpoint & make an arc. 3. Take your compass and place the tip on the other endpoint & make an arc that intersects your 1st arc. 4. Make a point where your two arcs intersect. Connect your 3 points. C Construction of an Equilateral Triangle Try this! Create an equilateral triangle using the segment below. 11) Construction of an Equilateral Triangle Try this! Create an equilateral triangle using the segment below. 12) Extension: Construction of an Equilateral Triangle w/ rod, string & pencil 1. Construct a segment of any length. 3. Switch the places of your rod & pencil tip. Make an arc that intersects your 1st arc. 2. Place your rod on one endpoint & the pencil on the other. Extend your pencil to make an arc. 4. Make a point where your two arcs intersect. Connect your 3 points. C Construction of an Equilateral Triangle w/ rod, string & pencil Try this! Create an equilateral triangle using the segment below. 13) Construction of an Equilateral Triangle w/ rod, string & pencil Try this! Create an equilateral triangle using the segment below. 14) Video Demonstrating Constructing Equilateral Triangles using Dynamic Geometric Software Click here to see video Angles & Angle Addition Postulate Identifying Angles An angle is formed by two rays with a common endpoint (vertex) The angle shown can be called , , or . C When there is no chance of confusion, the angle may also be identified by its vertex B. (Side) 32° The sides of are rays BC and BA B (Vertex) (Side) The measure of the angle is 32 degrees. "The measure of is equal to the measure of ..." A Congruent angles Two angles that have the same measure are congruent angles. The single mark through the arc shows that the angle measures are equal Exterior Interior We read this as is congruent to The area between the rays that form an angle is called the interior. The exterior is the area outside the angle. Constructing Congruent angles Given: <FGH 1. Draw a reference line w/ your straight edge. Place a reference point to indicate where your new segment starts on the line. 2. Place your compass point on the vertex (point G). 4. Swing an arc with the pencil so that it crosses both sides of <FGH. 3. Stretch the compass to any length so long as it stays ON the angle. Constructing Congruent angles (cont'd) 5. Without changing the span of the compass, place the compass tip on your reference point & swing an arc that goes through the line & extends above the line. 6. Go back to <FGH & measure the width of the arc from where it crosses one side of the angle to where it crosses the other side of the angle. The width of the arc is 2.2 cm Constructing Congruent angles (cont'd) 7. With this width, place the compass point on the reference line where your new arc crosses the reference line. Mark off this width on your new arc. The width of the arc is 2.2 cm 8. Connect this new intersection point to the starting point on the reference line. Constructing Congruent angles Try this! Construct a congruent angle on the given line. 15) Constructing Congruent angles Try this! Construct a congruent angle on the given line. 16) Video Demonstrating Constructing Congruent Angles using Dynamic Geometric Software Click here to see video Angle Measures Angles are measured in degrees, using a protractor. Every angle has a measure from 0 to 180 degrees. Angles can be drawn any size, the measure would still be the same. D A C B is a 23° degree angle The measure of is 23° degrees is a 119° degree angle The measure of is 119° degrees In and , notice that the vertex is written in between the sides M L Example K N O P J 90o clic 32o clic 180o clic k k 63o clic k k 135o clic k L M Example K N O P J Challenge Questions clic k 148 -117 31o clic k 117 - 90 27o clic k 90 -45 45o clic k 45o Angle Relationships Once we know the measurements of angles, we can categorize them into several groups of angles: 0° < acute < 90° right = 90° 90° < obtuse < 180° straight = 180° 180° 180° < reflex angle < 360° Two lines or line segments that meet at a right angle are said to be perpendicular. Click here for a Math Is Fun website activity. Adjacent Angles O K Adjacent angles are angles that have a common ray coming out of the vertex going between two other rays. P In other words, they are angles that are side by side, or adjacent. J Angle Addition Postulate if a point S lies in the interior of then PQS + SQR = PQR. PQR, 58° 32° 26° m PQS = 32° + m SQR = 26° m PQR = 58° Just as from the Segment Addition Postulate, "The whole is the sum of the parts" Example 72 Given m∠ABC = 22° and m∠DBC = 46°. Answer Find m∠ABD Hint: Always label your diagram with the information given click Given m∠OLM = 64° and m∠OLN = 53°. Find m∠NLM A 28 B 15 C 11 D 117 64° 53° Answer 73 Given m∠ABD = 95° and m∠CBA = 48°. Find m∠DBC Answer 74 Given m∠KLJ = 145° and m∠KLH = 61°. Find m∠HLJ Answer 75 Given m∠TRQ = 61° and m∠SRQ = 153°. Find m∠SRT Answer 76 C is in the interior of ∠TUV. If m∠TUV = (10x + 72)⁰, m∠TUC = (14x + 18)⁰ and m∠CUV = (9x + 2)⁰ solve for x. Answer 77 Hint: Draw a diagram and label it with the given information click D is in the interior of ∠ABC. If m∠CBA = (11x + 66)⁰, m∠DBA = (5x + 3)⁰ and m∠CBD= (13x + 7)⁰ solve for x. Answer 78 Hint: Draw a diagram and label it with the given information click F is in the interior of ∠DQP. If m∠DQP = (3x + 44)⁰, m∠FQP = (8x + 3)⁰ and m∠DQF= (5x + 1)⁰ solve for x Answer 79 Angle Pair Relationships Complementary Angles A pair of angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the angles is said to be the complement of the other. These two angles are complementary (58° + 32° = 90°) We can rearrange the angles so they are adjacent, i.e. share a common side and a vertex. Complementary angles do not have to be adjacent. If two adjacent angles are complementary, they form a right angle Supplementary Angles Supplementary angles are pairs of angles whose measurements sum to 180 degrees. Supplementary angles do not have to be adjacent or on the same line; they can be separated in space. One angle is said to be the supplement of the other. K 136 degrees 44 degrees P O J Example Solution: Choose a variable for the angle - I'll choose "x" Example Two angles are complementary. The larger angle is twice the size of the smaller angle. What is the measure of both angles? Let x = the angle Since the angles are complementary we know their sum must equal 90 degrees. 90 = 2x + x 90 = 3x 30 = x 80 An angle is 34° more than its complement. Answer What is its measure? Hint: Choose a variable for the angle. What is a complement? click 81 An angle is 14° less than its complement. Answer What is the angle's measure? Hint: Choose a variable for the angle. What is a complement? click 82 An angle is 98 more than its supplement. Answer What is the measure of the angle? Hint: Choose a variable for the angle What is a supplement? click An angle is 74° less than its supplement. What is the angle? Answer 83 An angle is 26° more than its supplement. What is the angle? Answer 84 Linear Pair of Angles If the two supplementary angles are adjacent, having a common vertex and sharing one side, their non-shared sides form a line. A linear pair of angles are two adjacent angles whose non- common sides on the same line. A line could also be called a straight angle with 180° Vertical Angles Given: m∠ABC = 55o Find the measures of the remaining angles. A 55o B E D C Answer Vertical Angles: Two angles whose sides form two pairs of opposite rays - In diagram below, ∠ABC & ∠DBE are vertical angles, and ∠ABE & ∠CBD are vertical angles. Vertical Angles Given: m∠ABC = 55o What do you notice about your vertical angles? A 125o E 55o 55o B 125o C D click to reveal theorem Vertical Angles Theorem: All vertical angles are congruent. Example Find the m < 1, m < 2 & m < 3. Explain your answer. 1 36o 3 2 36 + m < 1 = 180 -36 m < 1 = 144o Linear pair angles are supplementary m < 2 = 36o; Vertical angles are congruent (original angle & < 2) m < 3 = 144o; Vertical angles are congruent (< 1 & < 3) -36 What is the m < 1? A 77o 1 2 B 103o C 113o D none of the above 77o 3 Answer 85 What is the m < 2? A 77o 1 2 B 103o C 113o D none of the above 77o 3 Answer 86 What is the m < 3? A 77o 1 2 B 103o C 113o D none of the above 77o 3 Answer 87 What is the m < 4? A 112o 4 112o 6 B 78o C 102o D none of the above 5 Answer 88 What is the m < 5? A 112o 4 112o 6 B 68o C 102o D none of the above 5 Answer 89 What is the m < 6? A 102o 4 112o 6 B 78o C 112o D none of the above 5 Answer 90 Example Find the value of x. The angles shown are vertical angles, so they are congruent (13x + 16)o (14x + 7)o 13x + 16 = 14x + 7 -13x -13x 16 = x + 7 -7 -7 9=x Example Find the value of x. The angles shown are a linear pair, so they are supplementary (2x + 8)o (3x + 17)o 2x + 8 + 3x + 17 = 180 5x + 25 = 180 - 25 5x = 155 5 x = 31 - 25 5 Find the value of x. A 95 B 50 C 45 D 85o (2x - 5)o 40 Answer 91 92 Find the value of x. A 75 75o (6x + 3)o B 17 C 13 12 Answer D 93 Find the value of x. A 122o 13.1 (9x - 4)o B 14 C 15 122 Answer D Find the value of x. A 12 (7x + 54)o B 13 C 42 D 42o 138 Answer 94 Angle Bisectors & Constructions Angle Bisector bisects An angle bisector is a ray or line which starts at the vertex and cuts an angle into two equal halves Bisect means to cut it into two equal parts. The 'bisector' is the thing doing the cutting. The angle bisector is equidistant from the sides of the angle when measured along a segment perpendicular to the sides of the angle. Constructing Angle Bisectors 1. With the compass point on the vertex, draw an arc across each side 3. With a straightedge, connect the vertex to the arc intersections 2. Without changing the compass setting, place the compass point on the arc intersections of the sides 4. Label your point X m∠UVX = m∠WVX ∠UVX ∠WVX Try this! Bisect the angle 17) Try this! Bisect the angle 18) Constructing Angle Bisectors w/ string, rod, pencil & straightedge 1. With the rod on the vertex, draw an arc across each side. 3. With a straightedge, connect the vertex to the arc intersections 2. Place the rod on the arc intersections of the sides & draw 2 arcs, one from each side showing an intersection point. 4. Label your point X m∠UVX = m∠WVX ∠UVX ∠WVX Try this! Bisect the angle with string, rod, pencil & straightedge. 19) Try this! Bisect the angle with string, rod, pencil & straightedge. 20) Constructing Angle Bisectors by Folding 1. On patty paper, create any angle of your 2. Fold your patty paper so that BA choice. Make it appear large on your patty lines up with BC. Crease the fold. paper. Label the points A, B & C 3. Unfold your patty paper. Draw a ray along the fold, starting at point B. 4. Draw & label your point. Try this! 21) Bisect the angle with folding. Try this! 22) Bisect the angle with folding. Videos Demonstrating Constructing Angle Bisectors using Dynamic Geometric Software Click here to see video using a compass and segment tool Click here to see video using the menu options Finding the missing measurement. Answer Example: ABC is bisected by BD. Find the measures of the missing angles. A D 52o B C Example: EFG is bisected by FH. The m EFG = 56o. Find the measures of the missing angles. Answer E H 56o F G Example: MO bisects LMN. Find the value of x. L (x + 10)o (3x - 20)o N O Answer M JK bisects HJL. Given that m HJL = 46o, what is m HJK? Answer 95 Hint: click to does reveal bisect mean? What Draw & label a picture. NP bisects MNO Given that m MNP = 57o, what is m MNO? Answer 96 Hint: What does bisect mean? click to reveal Draw & label a picture. RT bisects QRS Given that m QRT = 78o, what is m QRS? Answer 97 VY bisects UVW. Given that m UVW = 165o, what is m UVY? Answer 98 99 BD bisects ABC. Find the value of x. A D Answer (7x + 3)o (11x - 25)o B C 100 FH bisects EFG. Find the value of x. H E Answer (9x - 17)o (3x + 49)o F G JL bisects IJK. Find the value of x. I L (7x + 1)o (12x - 19)o J K Answer 101