1. Unit 3 Triangles

2. Chapter Objectives Classification of Triangles by Sides Classification of Triangles by Angles Exterior Angle Theorem Triangle Sum Theorem Adjacent Sides and Angles Parts of Specific Triangles 5 Congruence Theorems for Triangles

3. Lesson 3.1 Classifying Triangles

4. Lesson 3.1 Objectives Classify triangles according to their side lengths. (G1.2.1) Classify triangles according to their angle measures. (G1.2.1) Find a missing angle using the Triangle Sum Theorem. (G1.2.2) Find a missing angle using the Exterior Angle Theorem. (G1.2.2)

5. Classification of Triangles by Sides NameEquilateralIsoscelesScalene Looks Like Characteristics3 congruent sides At least 2 congruent sides No Congruent Sides

6. Classification of Triangles by Angles NameAcuteEquiangularRightObtuse Looks Like Characteristics3 acute angles 3 congruent angles 1 right angles 1 obtuse angle

7. Example 3.1 Classify the following triangles by their sides and their angles. Scalene Obtuse Scalene Right Isosceles Acute Equilateral Equiangular

8. Vertex The vertex of a triangle is any point at which two sides are joined. –It is a corner of a triangle. There are 3 in every triangle

9. Adjacent Sides and Adjacent Angles Adjacent sides are those sides that intersect at a common vertex of a polygon. –These are said to be adjacent to an angle. Adjacent angles are those angles that are right next to each other as you move inside a polygon. –These are said to be adjacent to a specific side.

10. More Parts of Triangles If you were to extend the sides you will see that more angles would be formed. So we need to keep them separate –The three angles are called interior angles because they are inside the triangle. –The three new angles are called exterior angles because they lie outside the triangle.

11. Theorem 4.1: Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180 o. A B C m  A + m  B + m  C = 180 o

12. Example 3.2 Solve for x and then classify the triangle based on its angles. 3x + 2x + 55 = 180 Triangle Sum Theorem 5x + 55 = 180 Simplify 5x = 125 SPOE x = 25 DPOE Acute 75 o 50 o

13. Example 3.3 Solve for x and classify each triangle by angle measure. 1. Right 2. Acute

14. Example 3.4 Draw a sketch of the triangle described. Mark the triangle with symbols to indicate the necessary information. 1.Acute Isosceles 2.Equilateral 3.Right Scalene

15. Example 3.5 Draw a sketch of the triangle described. Mark the triangle with specific angle measures, side lengths, or symbols to indicate the necessary information. 1.Obtuse Scalene 2.Right Isosceles 3.Right Equilateral (Not Possible)

16. Theorem 4.2: Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. m  A +m  B = m  C A B C

17. Example 3.6 Solve for x Exterior Angles Theorem Combine Like Terms Subtraction Property Addition Property Division Property

18. Corollary to the Triangle Sum Theorem A corollary to a theorem is a statement that can be proved easily using the original theorem itself. –This is treated just like a theorem or a postulate in proofs. The acute angles in a right triangle are complementary. C A B m  A + m  B = 90 o

19. Example 3.7 Find the unknown angle measures. 1. 2. 3. 4. VA If you don’t like the Exterior Angle Theorem, then find m  2 first using the Linear Pair Postulate. Then find m  1 using the Angle Sum Theorem. VA

20. Homework 3.1 Lesson 3.1 – All Sections –p1-6 Due Tomorrow

21. Lesson 3.2 Inequalities in One Triangle

22. Lesson 3.2 Objectives Order the angles in a triangle from smallest to largest based on given side lengths. (G1.2.2) Order the side lengths of a triangle from smallest to largest based on given angle measures. (G1.2.2) Utilize the Triangle Inequality Theorem.

23. Theorem 5.10: Side Lengths of a Triangle Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. –Basically, the larger the side, the larger the angle opposite that side. Longest side Largest Angle 2 nd Longest Side 2 nd Largest Angle Smallest Side Smallest Angle

24. Theorem 5.11: Angle Measures of a Triangle Theorem If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. –Basically, the larger the angle, the larger the side opposite that angle. Longest side Largest Angle 2 nd Longest Side 2 nd Largest Angle Smallest Side Smallest Angle

25. Example 3.8 Order the angles from largest to smallest. 1. 2. 3.

26. Example 3.9 Order the sides from largest to smallest. 1. 2. 33 o

27. Example 3.10 Order the angles from largest to smallest. 1.In  ABC AB = 12 BC = 11 AC = 5.8 Order the sides from largest to smallest. 2.In  XYZ m  X = 25 o m  Y = 33 o m  Z = 122 o

28. Theorem 5.13: Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Add each combination of two sides to make sure that they are longer than the third remaining side. 6 6 6 1 3 3 2 4 4

29. Example 3.11 Determine whether the following could be lengths of a triangle. a)6, 10, 15 a)6 + 10 > 15 10 + 15 > 6 6 + 15 > 10 YES! b)11, 16, 32 b)11 + 16 < 32 NO! Hint: A shortcut is to make sure that the sum of the two smallest sides is bigger than the third side. The other sums will always work.

30. Homework 3.2 Lesson 3.2 – Inequalities in One Triangle –p7-8 Due Tomorrow Quiz Friday, October 15 th

31. Lesson 3.3 Isosceles, Equilateral, and Right Triangles

32. Lesson 3.3 Objectives Utilize the Base Angles Theorem to solve for angle measures. (G1.2.2) Utilize the Converse of the Base Angles Theorem to solve for side lengths. (G1.2.2) Identify properties of equilateral triangles to solve for side lengths and angle measures. (G1.2.2)

33. Special Parts of an Isosceles Triangle An isosceles triangle has only two congruent sides –Those two congruent sides are called legs. –The third side is called the base. legs base

34. Isosceles Triangle Theorems Theorem 4.6: Base Angles Theorem –If two sides of a triangle are congruent, then the angles opposite them are congruent. Theorem 4.7: Converse of Base Angles Theorem –If two angles of a triangle are congruent, then the sides opposite them are congruent.

35. Example 3.12 Solve for x and y. 1. 2. 3. 4. 75 o 55 o 5. = 90 o + 45 o = 4545 =

36. Equilateral Triangles Corollary to Theorem 4.6 –If a triangle is equilateral, then it is equiangular. Corollary to Theorem 4.7 –If a triangle is equiangular, then it is equilateral.

37. Example 3.13 Solve for x and y. 1. Or…In order for a triangle to be equiangular, all angles must equal… It does not matter which two sides you set equal to each other, just pick the pair that looks the easiest! 5x o 2.

38. Special Parts in a Right Triangle Right triangles have special names that go with its parts as well. For instance: –The two sides that form the right angle are called the legs of the right triangle. –The side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side of a right triangle. legs hypotenuse

39. Homework 3.3 Lesson 3.3 – Isosceles, Equilateral, and Right Triangles –p9-11 Due Tomorrow Quiz Tomorrow –Tuesday, October 19 th

40. Lesson 3.4 MediansAnd Altitudes of Triangles

41. Lesson 3.4 Objectives Identify a median, an altitude, and a perpendicular bisector of a triangle. (G1.2.5) Identify a centroid of a triangle. Utilize medians and altitudes to solve for missing parts of a triangle. (G1.2.5) Identify the orthocenter of a triangle.

42. Perpendicular Bisector A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called the perpendicular bisector.

43. Triangle Medians A median of a triangle is a segment that does the following: –Contains one endpoint at a vertex of the triangle, and –Contains its other endpoint at the midpoint of the opposite side of the triangle. A B C D

44. Centroid When all three medians are drawn in, they intersect to form the centroid of a triangle. –This forms a point of concurrency which is defined as a point formed by the intersection of two or more lines. The centroid happens to find the balance point for any triangle. In Physics, this is how we locate the center of mass. AcuteRight Obtuse Remember: All medians intersect the midpoint of the opposite side.

45. Theorem 5.7: Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side. –The centroid is 2 / 3 the distance from any vertex to the opposite side. Or said another way, the centroid is twice as far away from the opposite angle as it is to the nearest side. AP = 2 / 3 AE BP = 2 / 3 BF CP = 2 / 3 CD 2 / 3 AE 2 / 3 BF 2 / 3 CD

46. Example 3.14 S is the centroid of  RTW, RS = 4, VW = 6, and TV = 9. Find the following: a)RV a)6 b)RU b)6 Half of 4 is 2, and … 4 + 2 = 6 Works every time!! c)SU c)2 d)RW d)12 e)TS e)6 6 is 2 / 3 of 9 f)SV f)3 Half of 6, which is the other part of the median.

47. Altitudes An altitude of a triangle is the perpendicular segment from a vertex to the opposite side. –It does not bisect the angle. –It does not bisect the side. The altitude is often thought of as the height. –While true, there are 3 altitudes in every triangle but only 1 height!

48. Orthocenter The three altitudes of a triangle intersect at a point that we call the orthocenter of the triangle. The orthocenter can be located: –inside the triangle –outside the triangle, or –on one side of the triangle Acute Right Obtuse The orthocenter of a right triangle will always be located at the vertex that forms the right angle.

49. Example 3.15 Is segment BD a median, altitude, or perpendicular bisector of  ABC? Hint: It could be more than one! 1. Median Altitude PerpendicularBisector None 2. 3. 4. Median None

50. Homework 3.4 Lesson 3.4 – Altitudes and Medians –p12-13 Due Tomorrow

51. Lesson 3.5 Area and Perimeter of Triangles

52. Lesson 3.5 Objectives Find the perimeter and area of triangles. (G1.2.2)

53. Reviewing Altitudes Determine the size of the altitudes of the following triangles. I. 6 II. 16 III. ? If it is a right triangle, then you can use Pythagorean Theorem to solve for the missing side length.

54. Area The area of a figure is defined as “the amount of space inside the boundary of a flat (2-dimensional) object” –http://www.mathsisfun.com/definitions/area.html Because of the 2-dimensional nature, the units to measure area will always be “squared.” –For example: in 2 or square inches ft 2 or square feet m 2 or square meters mi 2 or square miles The area of a rectangle has up until now been found by taking: length x width (l x w) We will now change the wording slightly to fit a more general pattern for all shapes, and that is: base x height (b x h) That general pattern will exist as long as the base and height form a right angle. –Or said another way, the base and height both touch the right angle. w l h b

55. Area of a Triangle The area of a triangle is found by taking one-half the base times the height of the triangle Again the base and height form a right angle. –Notice that the base is an actual side of the triangle, and… –The height is nothing more than the altitude of the triangle drawn from the base to the opposite vertex. h b h b

56. Perimeter of a Triangle The perimeter of a triangle is found by taking the sum of all three sides of the triangle. –So basically you need to add all three sides together. The perimeter is a 1-dimensional measurement, so the units should not have an exponent on them. –Example: »in »ft »m»m »mi c h a b

57. Example 3.16 Find the area and perimeter of the following triangles. 1.

58. Homework 3.5 Lesson 3.5 – Area and Perimeter of Triangles –p14-15 Due Tomorrow