2. TheoremIfThen If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half the distance. D is the midpoint of CA and E is the midpoint of CB AB C DE // AB C DE 2x x

3.  Find all missing side lengths, given AB = 10, BE = 7, and DE = 4 AD = DB = EF = AF = FC = AC = EC = DF = BC = B A C D E F 10 7 4 5 5 5 4 4 8 7 7 14

4.  Solve for x if the perimeter of triangle ABC is 42 B A C D E F 12 3x – 3 9 3x - 3

5.  P. 288 #’s 10-18 even, 19-27, 29-32, 37, 40- 44, 49-52

7. TheoremIfThen If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. AB M P AB M P

8. TheoremIfThen If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment. AB P AB P M

9.  Calculate AC A BC D 6x – 104x Complete Got it 1, p. 293Ans: QR = 8

10.  The distance from a point to a line is the length of the perpendicular segment from the point to the line. B A l

11. TheoremIfThen If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. Q P S R Q P S R

12. TheoremIfThen 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. Q P S R Q P S R

13.  Calculate RM Complete Got it 3, p. 295Ans: FB = 21 A R P M 7x 2x+25

14.  P. 296 #’s 6-8, 12-22, 36, 39-46

16. TheoremDiagramSymbols Perpendicular bisectors are concurrent at P. The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices

17.  When 3 or more lines intersect at one point, they are concurrent. The point at which they intersect is the point of concurrency.  The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle.

18.  Since the circumcenter is equidistant from the vertices, you can use the circumcenter as the center of the circle that contains each vertex of the triangle. You say the circle is circumscribed about the triangle.

19.  Find the circumcenter of the following 3 points. Circumcenter

20. TheoremDiagramSymbols Angle bisectors are concurrent at P. The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.

21.  The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle.  P is the center of the circle that is inscribed in the triangle.

22.  If P is the incenter and PY = 2x-4 and PZ = 5x-13, find BP, if PB = 3x. 2x-4 5x-13 3x

23.  P. 305 #’s 8-12 even, 15-18, 22, 33-34, 37-40

25.  A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side. Median

26. TheoremDiagramSymbols The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.

27.  In a triangle, the point of concurrency of the medians is the centroid of the triangle. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance. For any triangle, the centriod is always inside the triangle.

28.  In the diagram below CE = 20, CH = 6, and DC = 8. Find CG, FC and DJ.

30. Complete Got it 1, p. 310Ans: a) ZC = 13.5 b) 2 : 1

31.  An altitude of a triangle is the perpendicular segment from the vertex of the triangle to the line containing the opposite side. An altitude of a triangle can be inside or outside the triangle, or it can be a side of the triangle.

32. TheoremDiagram The lines that contain the altitudes of a triangle are concurrent.

33.  The lines that contain the altitudes of a triangle are concurrent at the orthocenter of the triangle. The orthocenter can be inside, on, or outside the triangle. Acute Triangle Right Triangle Obtuse Triangle

34.  Triangle ABC has vertices A (1,3), B (2,7), and C (6,3). What are the coordinates of the orthocenter of the triangle? Graph the pointsCreate two altitudes Locate the point of intersection The orthocenter is the point (2, 4) Complete Got it 3, p. 311 Ans: (1, 2)

36.  P. 312 #’s 8-20, 24-27, 31, 37-41, 43-44

38. Comparison Property of Inequality b c a

39. CorollaryIfThen The measure of an exterior angle of a triangle is greater than the measure of each of the remote interior angles 1 2 3

40. Theorem 5-10IfThen If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. Theorem 5-11IfThen If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.

41.  In the diagram, list the sides in order from smallest to largest. Complete Got it 3, p. 327Ans: OX 30° 110°

42. TheoremDiagram The sum of the lengths of any two sides of a triangle is greater than the third side.

43.  Can a triangle have side lengths of 4, 4, and 8? How about 4, 5, and 6? Complete Got it 4, p. 327 Ans: a) No b) Yes No, a triangle cannot be made with side lengths of 4, 4, and 8. Yes, a triangle can be made with side lengths of 4, 5, and 6. The sum of the lengths of any two sides is greater than the third.

44.  If two sides of a triangle are 8 and 14, what is the range of possible lengths for the third side? Complete Got it 5, p. 328 Ans: 3 < X < 11 The length of a side of a triangle can never be negative so we can eliminate that answer. So the range of values is between 6 and 22.

45.  P. 328 #’s 6-34 even, 37-39, 43, 44, 46-49, 53-55.

47. Theorem 5-13IfThen If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third leg is opposite the larger included angle.

48. Theorem 5-14IfThen If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.

49.  What is the range of possible values for x? Complete Got it 3, p. 335Ans: -6 < x < 24 60° 15 10 (5x-20)° Find the upper limit Find the lower limit

50.  P. 336 #’s 6-9, 11-14, 16-18, 21, 24, 26-28, 31-35, 37-39