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Transcript
Unit 2 – Properties of Angles and Triangles By the end of this unit, I should be able to: Generalize the relationship between pairs of angles made by transversals and parallel lines Prove the properties of angles formed by transversals and parallel lines Find the relationship between the sum of the interior angles and the number of sides of a polygon Identify and correct errors in a proof Find and justify the measures in a diagram that involves parallel lines, angles and triangles Solve a contextual problem that involves triangles and angles Make parallel lines with only a compass or a protractor Determine whether lines are parallel given the measure of an angle at a transversal Getting started activity: Polygon quadrilateral trapezoid parallelogram rhombus rectangle square triangle scalene triangle isosceles triangle equilateral triangle acute triangle Properties Polygon in Dog obtuse triangle right triangle 2.1 Exploring Parallel Lines Important terminology for this unit: transversal: interior angles: exterior angles: corresponding angles: converse: 2.2 Angles Formed by Parallel Lines Remember these important facts before we go any further: If a transversal intersects two parallel lines, the corresponding angles are equal Two angles that form a straight line are supplementary Two angles whose sum is 180 are supplementary Vertically opposite angles are equal Transitive property: if a = b and a = c , then b = c How can you use a ruler and a compass to draw lines that are parallel? Write your steps and the parallel lines here: alternate interior angles: alternate exterior angles: Example 1 Reasoning about conjectures involving angles formed by transversals Make a conjecture that involves interior angles formed by parallel lines and a transversal. Prove your conjecture Statement Justification Try it: Make a conjecture that alternate exterior angles are equal. Prove your conjecture Statement Justification Example 2 Determining Measures Determine the measures of a, b, c and d b a c 60o d Example 3 One side of a cellphone tower will be built as shown. Use the angle measures to proves that braces CG, BF, and AE are parallel Statement Justification 2.3 Angles - Properties in Triangles Terminology: non-adjacent interior angles: Activity: On page 86, complete the “EXPLORE” activity with a partner. Provide a sketch of your paper below for reference: What do you notice about the three triangles? Can you use the angle relationship to show that the sum of the measures of the angles in any acute triangle formed this way is 180? Investigate: (p. 86): Can you prove that the sum of the measures of the interior angles of any triangle is 180? A. . B. Identify each pair of equal angles in your diagram. Explain how you know that the measures of the angles in each pair are equal. C. What is the sum of the measure of PDR, RDE and QDE? Explain how you know. D. Explain why: DRE + RED = 180 Example 1: In the diagram, MTH is an exterior angle of MAT Try:If you are given one interior angle and one exterior angle of a triangle, can you always determine the other interior angles of the triangle? Explain, using diagrams. Example #2: Determine the relationship between an exterior angle of a triangle and its non-adjacent interior angles. Try: Prove e = a + b Statement Justification Example 3: Determine the measures of NMO, MNO and QMO. Try: QP is parallel to MR. Determine the measure of MQO, MOQ, NOP, OPN and RNP 2.4 Angles – Properties in polygons Terminology: convex polygon: convex polygon: non-convex polygon (concave) EXPLORE: A pentagon has three right angles and four sides of equal length, and four sides of equal length. What is the sum of the measures of the angles in the pentagon? Investigate: How is the number of sides in a polygon related to the sum of its interior angles and the sum of its exterior angles? Refer to pg. 94 and follow the steps and answer the select questions. Polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Number of Sides Number of Triangles Sum of Angles D. Make a conjecture about the relationship between the sum of the measure of the interior angles of a polygon, S, and the number of the sides of the polygon, n. E. Use your conjecture to predict the sum of the measure of the interior angles of a dodecagon (12 sides). Verify your prediction using triangles. F. Determine the sum of the measures of the exterior angles. G. What do you notice about the sum of the measures of each exterior angle of your rectangle above and its adjacent interior angle? Would this relationship also hold for the exterior and interior angles of irregular quadrilateral shown? Explain. H. Make a conjecture about the sum of the measures of the exterior angles of any quadrilateral. Example #1: Prove that the sum of the measures of the interior angles of any n-sided convex polygon can be expressed as: 180o(𝑛 − 2) Example #2: Outdoor furniture and structures like gazebos sometimes use a regular hexagon in their building plan. Determine the measure of each interior angle of a regular hexagon. Your Turn: Determine the measure of each interior angle of a regular 15-sided polygon (a pentadecagon). Example #3: A floor tiler designs custom floors using tiles in the shape of regular polygons. Can the tiler use congruent regular octagons and congruent squares to tile a floor, if they have the same side length? Your Turn: Can a tiling pattern be created using regular hexagons and equilateral triangles that have the same side length? Explain.