1. Warm-up HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?

2. KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing? Answer:  A and  D

3. Unit 4 Lesson 4 Special Segments in Triangles and their intersections

4. Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. a. all angles whose measures are less than m  4 b. all angles whose measures are greater than m  8 Answer:  5,  2,  8,  7 Answer:  4,  9,  5

5. Perpendicular Bisector RECALL

6. Any point on the perpendicular bisector is the same distance from the endpoints of the segment. Same Distance

7. RECALL: Theorems Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

8. Perpendicular Bisector of a triangle: A perpendicular bisector of a side of a triangle is a line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side.

9. Vocabulary When three or more lines intersect at a common point, the lines are called concurrent lines and their point of intersection is call the point of concurrency. The point of concurrency of the perpendicular bisectors of triangle is called circumcenter.

10. Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle.

11. In a triangle there are three perpendicular bisectors. Where they meet is called the… Circumcenter The circumcenter is the same distance to each of the vertices.

12. Example A.–5 B.0.5 C.5 D.10 In the figure, A is the circumcenter of ΔLMN. Find y if LO = 8y + 9 and ON = 12y – 11.

13. Example 2 A.13 B.11 C.7 D.–13 In the figure, A is the circumcenter of ΔLMN. Find x if m  APM = 7x + 13.

14. You Try A.–12.5 B.2.5 C.10.25 D.12.5 In the figure, A is the circumcenter of ΔLMN. Find r if AN = 4r – 8 and AM = 3(2r – 11).

15. Vocabulary The angle bisectors of a triangle are congruent, and their point of concurrency is called the incenter of a triangle.

16. Angle Bisector If you have an angle bisector in a triangle it is the same distance from both the sides of the angle.

17. A triangle has three angle bisectors. Where they meet is called the… Incenter

18. Example 1-1a Given: Find the measure of angle DGE Answer: Use Angle Bisectors

19. Example 1-1b Answer: Given: Find the measure of angle ADC. Try it!

20. Example In the figure, point D is the incenter of ΔABC. What segment is congruent to DG? ___ A.DE B.DA C.DC D.DB ___

21. Example 2 A.  GCD B.  DCG C.  DFB D.  ADE In the figure, point D is the incenter of ΔABC. What angle is congruent to  DCF?

22. Vocabulary A median is a segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex. The point of concurrency for the medians is called the centriod.

23. FACT The centroid is the point of balance for any triangle.

24. .Centroid

25. Other then the centroid being the center of gravity for a triangle, there is something else special about the centroid. The centroid is always 2/3 the total distance from the vertex to the other side.

26. In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Centroid Theorem YV = 12 Simplify. Answer: YP = 8; PV = 4 YP + PV= YVSegment Addition 8 + PV= 12YP = 8 PV= 4Subtract 8 from each side.

27. Example 2 If G is the centriod, and GE = 6, how long is BE? A G F E D C B

28. You try A.LR = 15; RO = 15 B.LR = 20; RO = 10 C.LR = 17; RO = 13 D.LR = 18; RO = 12 In ΔLNP, R is the centroid and LO = 30. Find LR and RO.

29. New Vocabulary The altitude is a segment from the vertex of triangle to the opposite side that is perpendicular to that side. The intersection point of the altitudes of a triangle is called the orthocenter.

30. The everyday meaning of altitude… The vertical elevation of an object above a surface

31. Example 1-2a ALGEBRA Points U, V, and W are the midpoints of respectively. Find a, b, and c. Segment Measures

32. ALGEBRA Points T, H, and G are the midpoints of respectively. Find w, x, and y. Example 1-2b Answer: Try it!

33. Unfortunately, there is nothing special about the orthocenter, but you will still be required to find it.

34. Vocabulary Tree Map Point of Concurrency Perpendicular Bisectors Circumcenter Angle Bisectors IncenterMediansCentroidAltitudesOrthocenter

35. Classwork