1. Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude Perpendicular Bisector Angle Bisector

2. 2 Triangle Angle Sum Theorem The sum of the measures of the angles of a triangle is 180°. m ∠A + m ∠B + m ∠C = 180 A B C Ex: If m ∠A = 30 and m∠B = 70; what is m∠C ? m ∠ A + m ∠ B + m ∠ C = 180 30 + 70 + m∠C = 180 100 + m∠C = 180 m∠C = 180 – 100 = 80

3. Exterior Angle Theorem 1 2 34 P QR In the triangle below, recall that  1,  2, and  3 are _______ angles of ΔPQR. interior Angle 4 is called an _______ angle of ΔPQR. exterior An exterior angle of a triangle is an angle that forms a _________, (they add up to 180) with one of the angles of the triangle. linear pair ____________________ of a triangle are the two angles that do not form a linear pair with the exterior angle. Remote interior angles In ΔPQR,  1, and  2 are the remote interior angles with respect to  4. In ΔPQR,  4 is an exterior angle because  3 +  4 = 180. The measure of an exterior angle of a triangle is equal to sum of its ___________________ remote interior angles

4. Exterior Angle Theorem 1 2 345 In the figure, which angle is the exterior angle? 55 which angles are the remote the interior angles?  2 and  3 If  2 = 20 and  3 = 65, find  5 65 20 If  5 = 90 and  3 = 60, find  2 85 90 60 30

5. Exterior Angle Theorem

7. Inequalities Within a Triangle If the measures of three sides of a triangle are unequal, then the measures of the angles opposite those sides are unequal ________________. 13 8 11 L P M in the same order LP < PM <ML m  M < mPmPm  L <

8. Inequalities Within a Triangle If the measures of three angles of a triangle are unequal, then the measures of the sides opposite those angles are unequal ________________. in the same order JK < KW <WJ m  W < mKmKm  J < J 45° W K 60° 75°

9. In a right triangle, the hypotenuse is the side with the ________________. greatest measure WY > XW 3 5 4 Y W X WY > XY

10. Inequalities Within a Triangle The longest side is So, the largest angle is The largest angle is So, the longest side is

11. Triangle Inequality Theorem Triangle Inequality Theorem The sum of the measures of any two sides of a triangle is _______ than the measure of the third side. greater a b c a + b > c a + c > b b + c > a

12. Triangle Inequality Theorem Can 16, 10, and 5 be the measures of the sides of a triangle? No! 16 + 10 > 5 16 + 5 > 10 However, 10 + 5 > 16

13. Medians, Altitudes, Angle Bisectors Perpendicular Bisectors

14. Every triangle has 1. 3 medians, 2. 3 angle bisectors and 3. 3 altitudes.

15. A B C Given  ABC, identify the opposite side 1. of A. 2. of B. 3. of C. BC AC AB Just to make sure we are clear about what an opposite side is…..

16. A new term… Point of concurrency Where 3 or more lines intersect

17. Any triangle has three medians. B A C M N L Let L, M and N be the midpoints of AB, BC and AC respectively. CL, AM and NB are medians of  ABC. Definition of a Median of a Triangle A median of a triangle is a segment whose endpoints are a vertex and a midpoint of the opposite side

18. The point where all 3 medians intersect Centroid Is the point of concurrency

19. The centroid is 2/3’s of the distance from the vertex to the side. from the vertex to the side. 2x x 10 5 32 X16

20. The centroid is the center of balance for the triangle. You can balance a triangle on the tip of your pencil if you place the tip on the centroid

21. angle bisector of a triangle a segment that bisects an angle of the triangle and goes to the opposite side.

22. A B C D E F In the figure, AF, DB and EC are angle bisectors of  ABC. Any triangle has three angle bisectors. Note: An angle bisector and a median of a triangle are sometimes different. M Let M be the midpoint of AC. The median goes from the vertex to the midpoint of the opposite side. BM is a median BD is a angle bisector of  ABC.

23. The Incenter is where all 3 Angle bisectors intersect Incenter Is the point of concurency

24. Any point on an angle bisector is equidistance from both sides of the angle

25. This makes the Incenter an equidistance from all 3 sides

26. D Theorem: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Let AD be a bisector of  BAC, P lie on AD, PM  AB at M, NP  AC at N. A B C M N P Then P is equidistant from AB and AC.

27. Theorem: Theorem: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. The converse of this theorem is not always true. Theorem: Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then the point lies on the bisector of the angle.

28. Using the Angle Bisector Theorem What is the length of RM? Because angle N has been bisected, I know that each point along the bisector is equidistant to the sides Since MR and RP are both perpendicular to each side and touch the bisector, I know they are equal 7x = 2x + 25 5x = 25 x = 5

29.  What is the length of FB? Because angle C has been bisected, I know that each point along the bisector is equidistant to the sides Since BF and FD are both perpendicular to each side and touch the bisector, I know they are equal 6x +3 = 4x + 9 2x +3 = 9 2x = 6 x = 3

30. Any triangle has three altitudes. Definition of an Altitude of a Triangle A altitude of a triangle is a segment that has one endpoint at a vertex and the other creates a right angle at the opposite side. The altitude is perpendicular to the opposite side while going through the vertex ACUTEOBTUSE B A C

31. RIGHT A B C If  ABC is a right triangle, identify its altitudes. BG, AB and BC are its altitudes. G Can a side of a triangle be its altitude?YES!

32. Orthocenter is where all the altitudes intersect. Orthocenter

33. The orthocenter can be located in the triangle, on the triangle or outside the triangle. Right Legs are altitudes Obtuse

34. A Perpendicular bisector of a side does not have to start at a vertex. It will form a 90° angles and bisect the side. a 90° angles and bisect the side. Circumcenter Is the point of concurrency

35. Any point on the perpendicular bisector of a segment is equidistance from the endpoints of the segment. A B C D AB is the perpendicular bisector of CD

36. This makes the Circumcenter an equidistance from the 3 vertices

37. Perpendicular Bisector

39. Using the Perpendicular Bisector Theorem What is the length of AB? Since BD is a perpendicular bisector, I know that BA and BC are congruent since they are connected to the vertex and the end of the bisected line. 4x = 6x – 10 –2x = – 10 x = 5 Since BD perpendicular to the side opposite B and bisects AC, I know that BD is a perpendicular bisector. AB = 4x AB = 4(5) AB = 20 BC = 6x – 10 BC = 6(5) – 10 BC = 20

40. Since SQ is perpendicular to the side opposite Q and bisects PR, I know that SQ is a perpendicular bisector.  What is the length of QR? 3n – 1= 5n – 7Since SQ is a perpendicular bisector, I know that PQ and QR are congruent since they are connected to the vertex and the end of the bisected line. – 1= 2n – 7 6 = 2n 3 = n PQ = 3n – 1 PQ = 3(3) –1 PQ = 8 QR = 5(3) – 7 QR = 5(n) – 7 QR = 8

41. The Midsegment of a Triangle is a segment that connects the midpoints of two sides of the triangle. D B C E A D and E are midpoints DE is the midsegment The midsegment of a triangle is parallel to the third side and is half as long as that side.

42. Midsegment Theorem D B C E A

43. 1.Identify the 3 pairs of parallel lines shown above

44. 2a. 2b.

45. Example 1 In the diagram, ST and TU are midsegments of triangle PQR. Find PR and TU. PR = ________ TU = ________16 ft 5 ft

46. Example 2 In the diagram, XZ and ZY are midsegments of triangle LMN. Find MN and ZY. MN = ________ ZY = ________53 cm 14 cm

47. Example 3 In the diagram, ED and DF are midsegments of triangle ABC. Find DF and AB. DF = ________AB = ________2652 3X – 4 5X+2 x = ________10 2 (DF ) = AB 2 (3x – 4 ) = 5x + 2 6x – 8 = 5x + 2 x – 8 = 2 x = 10

48. Perpendicular Bisectors A point is equidistant from two objects if it is the same distance from each. Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

49. Angle Bisectors The distance from a point to a line is the length of the perpendicular segment from the point to the line. Angle Bisector Theorem: If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem: If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

50. There are 3 of each of these special segments in a triangle. segments in a triangle. The 3 segments are concurrent. They intersect at the same point. This point is called the point of concurrency. The points have special names and special properties.

51. Altitude.. Vertex.. 90°.. Orthocenter Vertex.. 90°.. Orthocenter Angle Bisector.. Angle into 2 equal angles.. Incenter Angle into 2 equal angles.. Incenter Perpendicular Bisector… 90°.. bisects side.. Circumcenter 90°.. bisects side.. Circumcenter Median.. Vertex.. Midpoint of side.. Centroid Vertex.. Midpoint of side.. Centroid

52. Give the best name for AB ABABABABAB || | | || Median Altitude None AngleBisector PerpendicularBisector

53. Survival Training You’re Stranded On A Triangular Shaped Island. The Rescue Ship Can Only Dock On One Side Of The Island But You Don’t Know Which Side. At Which Point Of Concurrency Would You Set Up Camp So You Are An Equal Distance From All 3 Sides? INCENTER

54. What If The Ship Could Only Dock At One Of The Vertices? Would You Change The Location Of Your Camp ? If So, Where? YES CIRCUMCENTER CIRCUMCENTER

55. Where would you place a fire hydrant to make it equidistance to the houses and equidistance to the streets? ELM POST

56. ELM POST Angle bisector for the streets Perpendicular bisector for houses