In this chapter students will:

1. MVUSD GeoMetry CH-2 Instructor: Leon Robert ManuelPRENTICE HALL MATHEMATICS: Tools for a Changing World  In this chapter students will: expand their understanding of basic geometric concepts by investigating shapes that occur in everyday life: polygons, lines, and circles classify triangles and other polygons connect geometry to algebra by relating parallel and perpendicular to the concept of slope explore angles and arcs of a circle and apply those ideas to circle graphs learn about congruence and similarity draw three-dimensional figures using isometric and orthographic drawings

2. Chapter 2Section 2.01 Instructor: Leon Robert Manuel CA Geometry STD Triangles Exploring the exterior angles of a polygon.

3. Congruent Figures GEOMETRY LESSON 2-1 Solve each equation. 1.x + 6 = 25 2.x + 7 + 13 = 33 3. 5x = 540 4.x + 10 = 2x 5. For the triangle at the right, use the Triangle Angle-Sum Theorem to find the value of y.

4. Warm Up Classify each angle as acute, obtuse, or right. 1.2. 3. 4. If the perimeter is 47, find x and the lengths of the three sides. right acute obtuse x = 5; 8; 16; 23

5. Objectives Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths.

6. Vocabulary acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle

7. Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths.

8. C A AB, BC, and AC are the sides of ABC. B A, B, C are the triangle's vertices.

9. Prerequisite Skills • solve one-variable equations • recognize supplementary and complimentary angles • apply the Triangle-Sum Theorem • recognize that division by zero is undefined • recognize equations of vertical and horizontal lines • use the distance formula • find slope • use the distance and midpoint formulas • find percents • write and solve proportions • find perimeter • draw different views of a cube

10. Triangles • Triangle is a term used to describe one of the basic shapes of geometry. • A polygon with three corners or vertices and three sides or edges which are straight line segments. • In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e. two dimensional Cartesian space.

11. Equilateral Triangle • In an equilateral triangle, all sides are of equal length. An equilateral triangle is also an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon

12. Isosceles Triangle • Isosceles triangle, two sides are of equal length. • An isosceles triangle also has two congruent angles (namely, the angles opposite the congruent sides). • An equilateral triangle is an isosceles triangle, but not all isosceles triangles are equilateral triangles.

13. Scalene Triangles • Scalene triangle, all sides have different lengths. The internal angles in a scalene triangle are all different.

14. Triangles can be classified according to the their internal angles. • A right triangle (or right-angled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse. • The hypotenuse is the longest side in the right triangle.

15. Obtuse Triangle • An obtuse triangle has one internal angle larger than 90° (an obtuse angle)

16. Acute Triangle • An acute triangle has internal angles that are all smaller than 90° (three acute angles). • An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles.

17. Triangle Facts • All Triangles are two-dimensional • The angles of a triangle add up to 180 degrees. An exterior angle of a triangle (an angle that is adjacent and supplementary to an internal angle) is always equal to the two angles of a triangle that it is not adjacent/supplementary to. • The triangle inequality theorem states that for any triangle, the measure of a given side must be less than or equal to the sum of the other two sides.

18. Remember! When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.

19. Simmilar Triangles • Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional.

20. Similar Triangles • Two triangles are similar if at least 2 corresponding angles are congruent. AAA • If two corresponding sides of two triangles are in proportion, and their included angles are congruent, the triangles are similar. SAS • If three sides of two triangles are in proportion, the triangles are similar. SSS

21. Congruent Triangles • For two triangles to be congruent, each of their corresponding angles and sides must be congruent.

22. Congruent Triangles • SAS Postulate: If two sides and the included angles of two triangles are correspondingly congruent, the two triangles are congruent. • SSS Postulate: If every side of two triangles are correspondingly congruent, the triangles are congruent.

23. Congruent Triangles • ASA Postulate: If two angles and the included sides of two triangles are correspondingly congruent, the two triangles are congruent. • AAS Theorem: If two angles and any side of two triangles are correspondingly congruent, the two triangles are congruent. • Hypotenuse-Leg Theorem: If the hypotenuses and 1 pair of legs of two right triangles are correspondingly congruent, the triangles are congruent.

24. Euclidean vs. Spherical Triangles • In Euclidean geometry, the sum of the internal angles of a triangle is equal to 180°. This allows determination of the third angle of any triangle as soon as two angles are known. • In spherical Geometry: The Meridians of longitude pass through both poles and are perpendicular to the equator.

25. Session Start Slide

26. Chapter 2 Section 2.02Instructor: Leon Robert Manuel CA Geometry STD Polygons Exploring equations of lines HW CH 2-2 Page 79-80 Even Problems.

27. Warm Up Classify each triangle by its angles and sides. 1. MNQ 2.NQP 3. MNP 4. Find the side lengths of the triangle. acute; equilateral obtuse; scalene acute; scalene 29; 29; 23

28. California Standards 12.0Students find and use measures of sides and of interior and exterior angles of triangles and polygonsto classify figures and solve problems. 13.0Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. HW CH 2-2 Page 79-80 Even Problems.

29. Triangle Angle Sum Theorem • The sum of the measures of the angles of a triangle is 180o. Review

30. Exterior Angle Theorem • The measure of each angle of a triangle equals the sum of the measures of its two remote interior angles. Add-on

31. Polygons • A polygon is a closed plane figure with at least three sides. • The sides meet at their end points and no adjacent sides are collinear.

32. Convex Polygons

33. Convex Polygons • A polygon is convex if no diagonal contains points outside the polygon. • In other words: a polygon is strictly convex if every line segment between two vertices of the polygon is strictly interior to the polygon except at its endpoints. • A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees.

34. Polygon Interior Angle Sum Theorem • The sum of the measures of the interior angles of an n-gon is:

35. Polygon Exterior angle-sum Theorem • Sum of the Measures of the exterior angles of a polygon, one at each vertex, is 360o.

36. Polygon Exterior angle-sum Theorem • Sum of the Measures of the exterior angles of a polygon, one at each vertex, is 360o.

37. Example-1 Using the Polygon Interior Angle-Sum Theorem for n = 5.

38. Example-1 Find the measure of an interior angle and an exterior angle of a regular octagon. Find the measure of an interior angle first. Sum of the measures of the interior angles(8-2)180 = 1080

39. Chinese Checkers (P-79) Find the measure of the acute triangles of each of the triangle.

40. Classify polygons by its number of sides. • Pentagon (5-sides) • Hexagon (6-sides) • Octagon (8-sides) • Decagon (10-sides) • Dodecagon (12-sides)

41. Concave Polygons • A polygon is concave if a diagonal contains points outside the polygon. • At least one internal angle of a concave polygon is larger than 180 degrees.

42. Session Start Slide

43. Warm Up Find the value of m. 1. 2. 3. 4. undefined 0

44. Chapter 2Section 2.03Instructor: Leon Robert Manuel CA Geometry STD Parallel and Perpendicular lines in the coordinate plane Writing Linear Equations HW CH 2-3 Page 86 Problems 122.

45. What is a Slope

46. Vocabulary rise run slope

47. The slopeof a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope.

49. AB Example 1A: Finding the Slope of a Line Use the slope formula to determine the slope of. Substitute (–2, 7) for (x1, y1) and (3, 7) for (x2, y2) in the slope formula and then simplify.

50. AC Example 1B: Finding the Slope of a Line Use the slope formula to determine the slope of . Substitute (–2, 7) for (x1, y1) and (4, 2) for (x2, y2) in the slope formula and then simplify.