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Transcript
MPM1DR UNIT #7 GETTING STARTED Definitions : 1) 2) 3) 4) 5) Polygon : A closed figure composed of straight line segments. Regular Polygon : A polygon with equal sides and angles. Congruent : A congruent polygon is identical in size and shape. Congruent means equal. Transversal : A line that intersects two or more other lines. Adjacent : Two angles or sides next to each other. Review of Shapes : Ex. Draw a detailed diagram for each of the following polygons. 1) Rectangle : 2) Square : 3) Parallelogram : 4) Rhombus : 5) Trapezoid : 6) Scalene Triangle : 7) Isosceles Triangle : 8) Equilateral Triangle : Straight Angles : * The sum of angles that form a straight angle is 180. a b a + b = 180 * Two angles whose sum is 180 are called supplementary angles. Page 1 of 13 MPM1DR Interior and Exterior Angles of a Triangle : * The sum of the interior angles in a triangle is 180. d f a c b e a + b + c = 180 * Each exterior angle equals the sum of the two interior angles opposite it. d = b + c and e = a + c and f = a + b * Also, each exterior angle is a supplementary angle with its adjacent interior angle. a + d = 180 and b + e = 180 and c + f = 180 Angle Properties of Parallel Lines : * When a transversal crosses two parallel lines, special angles are formed. a b c d g e f h transversal * Corresponding angles form an F pattern with the transversal and the parallel lines. Corresponding angles are equal. d = h and c = g and f = b and e = a * Alternate angles form a Z pattern with the transversal and the parallel lines. Alternate angles are equal. c = f and d = e * The interior angles on the same side of the transversal form a C pattern. The interior angles on the same side of the transversal are supplementary angles. d + f = 180 and c + e = 180 Page 2 of 13 MPM1DR Examples : 1. Use the angle properties to determine each unknown angle. a) b) x 74 41 95 x c) d) 50 78 x 45 x e) x f) x y 70 w 39 82 y z x z Page 3 of 13 MPM1DR g) 75 z x y 2. Solve for x to find each missing angle measure. a) b) y 3x - 6 130 x + 10 2x + 1 x + 5 x + 10 x x c) x + 9 2x Page 4 of 13 MPM1DR 7.1 EXPLORING INTERIOR ANGLES OF POLYGONS Definitions : 1) Vertex : A corner point of a polygon, where two sides meet. The plural form is vertices. 2) Diagonal : In a polygon, a line segment joining two vertices that are not next to each other (i.e. not joined by one side) Recall : The interior angles of a triangle always add up to 180. Maybe we can use this angle property to help us determine the sum of the interior angles of other polygons. Let’s draw non-intersecting diagonals to the divide the interior of the following polygons into non-overlapping triangles, then count the number of triangles. a) Quadrilateral b) Pentagon c) Hexagon d) Heptagon Ex. Use your findings to fill in the chart below. Polygon Number of Sides Triangle 3 Number of Triangles 1 Sum of Interior Angles 180 Quadrilateral Pentagon Hexagon Heptagon Octagon * The sum of interior angles of any polygon with n number of sides is : n 2 180 Page 5 of 13 MPM1DR Examples : 1. Determine the sum of the interior angles of the following polygons. a) b) 2. Calculate the measure of each interior angle of a regular polygon with 16 sides. 3. The sum of the interior angles of a polygon is 1620. How many sides does the polygon have? Page 6 of 13 MPM1DR 7.2 ANGLE PROPERTIES OF POLYGONS Definitions : 1) Convex Polygon : A polygon with every interior angle less than 180; any straight line through it crosses, at most, two sides. 2) Concave Polygon : A polygon with at least one interior angle greater than 180; a straight line through it may cross more than two sides. Examples : 1. Determine the sum of the exterior angles of the following convex polygons. a) A regular hexagon. b) 120 135 x 85 50 * The sum of the exterior angles of a convex polygon is 360. Examples : 2. Determine the measure of each exterior angle in a regular 15-sided polygon. Page 7 of 13 MPM1DR 3. Determine the measure of each missing angle and state the angle property used for each step. a) 55 b) 109 93 b a y z x 30 49 125 4. An interior angle of a parallelogram is the measure of the exterior angle adjacent to it multiplied by 2. Determine the measure of each interior angle and each exterior angle. Draw the parallelogram. 5. In ABC, the measure of B is 8 more than 3 times the measure of A. The measure of C is 20 less than four times the measure of A. Determine the measure of each interior and each exterior angle of ABC. Draw the triangle. Page 8 of 13 MPM1DR 7.3 EXPLORING QUADRILATERAL DIAGONAL PROPERTIES Definition : 1) Kite : A quadrilateral that has two pairs of equal sides with no parallel sides. Diaganal Properties : Type of Quadrilateral Square Diagonal Lengths Equal Diagonal Bisectors Angles formed by the Diagonals All 90 (Perpendicular to each other) Bisect each other Rhombus Not equal Bisect each other All 90 (Perpendicular to each other) Rectangle Equal Bisect each other Not 90 Opposite angles are equal (OAT), and adjacent angles add to 180 (SAT) Parallelogram Not equal Bisect each other Not 90 Opposite angles are equal (OAT), and adjacent angles add to 180 (SAT) Isosceles Trapezoid Equal Do not bisect each other, but they form two pairs of equal line segments Not 90 Opposite angles are equal (OAT), and adjacent angles add to 180 (SAT) Kite May or Only one is bisected may not be by the other equal Diagram All 90 (Perpendicular to each other) Examples : Page 9 of 13 MPM1DR 1. A square has three of its vertices at A(-2, 0), B(1, 4), and C(5, 1). Use diagonal properties to find the coordinates of the fourth vertex, D. Explain your method. 2. Hannah cut a quadrilateral from a piece of cardboard. The diagonals were congruent, perpendicular, and bisected each other. Which type of quadrilateral did Hannah cut out? 3. Describe the possible type(s) of quadrilaterals that could be made with each set of diagonals. Justify your answers. a) b) c) d) Page 10 of 13 MPM1DR 7.4 REASONING ABOUT TRIANGLE AND QUADRILATERAL PROPERTIES Definitions : 1) 2) 3) 4) 5) Midpoint : The point that divides a line segment into two equal parts. Midsegment : A line segment connecting the midpoints of two adjacent sides of a polygon. Median : The line drawn from a vertex of a triangle to the midpoint of the opposite side. Conjecture : A guess or prediction based on limited evidence. Counterexample : An example that proves that a hypothesis or conjecture is false. Examples : 1. Fill in the diagram for each of the following. (You will need a ruler.) a) Midpoint of side PQ. p b) Midsegment from AB and BC. A c) Median from vertex G. B G C R Q D I H You can form a conjecture about triangles and quadrilaterals and then test it using examples. You can use properties that you already know to help you. 2. Use examples to test the following conjecture. Conjecture : The shape formed by the midsegments of a quadrilateral has the same shape as the original quadrilateral. Page 11 of 13 MPM1DR 7.5 REASONING ABOUT PROPERTIES OF POLYGONS Definitions : 1) Bimedian : The line joining the midpoints of two opposite sides in a quadrilateral. 2) Centroid : The centre of an object’s mass; the point at which it balances. It is also known as the centre of gravity. 3) The centroid of a triangle is the intersection of its medians. 4) The centroid of a quadrilateral is the intersection of its bimedians. Examples : 1. Draw the median lines for the given triangle. Then locate its centroid. You will need a ruler. A B C 2. Draw the bimedians for the given quadrilateral. Then locate its centroid. You will need a ruler. P Q S R 3. Locate the centroid for the following quadrilaterals. Then draw the diagonals of the quadrilateral. What do you notice about the centroid and the intersection point of the diagonals? * The centroid of a square, rectangle, rhombus, and a parallelogram is the intersection point of the diagonals. Page 12 of 13 MPM1DR 4. Draw the bimedian that connects the non-parallel sides on the following trapezoids. a) Determine the length of the two parallel sides and of the bimedian. What do you notice? b) Determine whether or not the bimedian is parallel to the two parallel sides. * The bimedian of the non-parallel sides of the trapezoid is parallel to its bases. Also, its length is the mean (average) of the bases lengths. Page 13 of 13