1. GEOMETRY HELP Warm Up 1) What is the difference between a perpendicular bisector and an angle bisector? 2) True or False: A segment bisects another segment and creates four right triangles.

2. GEOMETRY HELP Paper Folding Bisectors page 272 baby blue section share a book with your partner 1) Make a conjecture about the bisectors of the angles of a triangle. 2) Make a conjecture about the perpendicular bisectors of the sides of a triangle.

3. GEOMETRY HELP Concurrent- When three or more lines intersect in one point Point of Concurrency- The point at which they intersect 3) Is there any relationship between the lengths of the point of concurrency and the vertex or sides of the triangle?

4. GEOMETRY HELP Theorems: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. (Gizmo Application )

5. GEOMETRY HELP Find the center of the circle that circumscribes XYZ. Concurrent Lines, Medians, and Altitudes LESSON 5-3

6. GEOMETRY HELP (continued) The lines y = 4 and x = 3 intersect at the point (3, 4). This point is the center of the circle that circumscribes XYZ. Quick Check Concurrent Lines, Medians, and Altitudes LESSON 5-3 Additional Examples Vocab: Circumcenter of a triangle Circumscribed about a triangle

7. GEOMETRY HELP Circumcenter of a Triangle- The point of concurrency of the perpendicular bisectors of a triangle Circle is circumscribed about the triangle. Incenter of a Triangle- The point of concurrency of the angle bisectors of a triangle We can inscribe a circle in the triangle.

8. GEOMETRY HELP Median of a Triangle Is a segment whose endpoints are a vertex and the midpoint of the opposite side. Are the medians of one triangle concurrent? If so, is there a relationship between the median and a point of concurrency? Centroid- Point of Concurrency of the medians

9. GEOMETRY HELP The centroid is the point of concurrency of the medians of a triangle. 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. (Theorem 5-8) WM = WXTheorem 5-8 2323 16 = WX Substitute 16 for WM. 2323 24 = WXMultiply each side by. 3232 M is the centroid of WOR, and WM = 16. Find WX. Because M is the centroid of WOR, WM = WX. 2323 Concurrent Lines, Medians, and Altitudes LESSON 5-3 Additional Examples Quick Check

10. GEOMETRY HELP Altitude of a Triangle is the perpendicular segment from a vertex to the line containing the opposite side. Theorem: The lines that contain the altitudes of a triangle are concurrent. In your notes, draw the altitude of a triangle for an Acute Triangle, Right Triangle and Obtuse Triangle.

11. GEOMETRY HELP Because LX = XM, point X is the midpoint of LM, and KX is a median of KLM. So KX is both a median and an altitude. Is KX a median, an altitude, neither, or both? Concurrent Lines, Medians, and Altitudes LESSON 5-3 Additional Examples Quick Check Because KX is perpendicular to LM at point X, KX is an altitude.

12. GEOMETRY HELP Practice 5-3 #1, 2, 5-11

13. GEOMETRY HELP Assignment P275-277 #1-4, 9, 12-16, 19-22

14. GEOMETRY HELP Find the relationship Create a completed triangle using straws. Try different length straws Find a relationship between the lengths of the sides that complete a triangle and that would not complete a triangle. What conclusion can you draw about the smallest angles and largest angles with respect to the sides?

15. GEOMETRY HELP What do we notice….. Triangle inequality Theorem. The sum of any two side lengths of a triangle is greater than the third side length. A BC AB + BC > AC BC + AC > AB AC + AB > BC

16. GEOMETRY HELP Examples Tell whether a triangle can have sides with the given lengths. Explain. 1.3, 5, 7 2. 4, 6, 11

17. GEOMETRY HELP Workbook Page 347

18. GEOMETRY HELP Comparison Property of Inequality If a = b + c and c > 0 then a > b Corollary to the Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles m m m<3 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

19. GEOMETRY HELP Assignment Due Monday, 11/28 Textbook Page 292 #1-27 odd Try to get this done in class, ideally your homework is to enjoy this break with family and friends!