1. 1. Given ABC is congruent to XYZ, AB = 30, BC = 45, AC = 40, XY = 5x + 10 and XZ = 3y + 7, find x and y. 2. Draw and label triangles JKL and XYZ. Indicate the additional pairs of corresponding parts to be proved by postulates. <J = <X and JK = XY by AAS. How to classify and measure triangles? How to identify angle relationships in triangles in order to measure? How to construct triangles?

2. 1 Marker Straightedge Protractor Compass Study Guide Construction Worksheet How to classify and measure triangles? How to identify angle relationships in triangles in order to measure? How to construct triangles?

3. Quiz 1: classify triangles Measure missing angles Transformation Quiz 2: Congruent triangles Congruence statements Isosceles triangles Equilateral triangles How to classify and measure triangles? How to identify angle relationships in triangles in order to measure? How to construct triangles?

4. Use ruler and protractor Use ruler and compass How to classify and measure triangles? How to identify angle relationships in triangles in order to measure? How to construct triangles?

5. Construct each 1. segment bisector 2. angle bisector 3. parallel lines 4. perpendicular lines Materials: Ruler Protractor

6. Chapter Five

7. To identify and use medians, altitudes, angle bisectors, and perpendicular bisectors in a triangle.

8. Define Concurrent Example Define Point of Concurrency Example When three or more lines have a point in common. The point of intersection of three or more lines.

9. What is the Angle Bisector Concurrency Conjecture? Example Define Incenter Example The three angle bisectors of a triangle are concurrent. The point of concurrency of the angle bisector.

10. What is the Perpendicular Bisector Conjecture? Example Define Circumcenter Example The three perpendicular bisectors of a triangle are concurrent. The point of concurrency of the perpendicular bisector.

11. What is the Altitude Concurrency Conjecture? Example Define Orthocenter Example The three altitude (or the lines containing the altitude) of a triangle are concurrent. The point of concurrency of the altitude.

12. What is the Circumcenter Conjecture? Example What is the incenter Conjecture? Example The circumcenter of a triangle is equidistant from the vertices. The incenter of a triangle is equidistant from the sides.

13. Define Circumscribed Circle Example Define inscribed Circle Example A circle formed around the exterior of a polygon. A circle formed around the interior of a polygon.

14. What is the Median Concurrency Conjecture? Example Define Centroid Example The three medians of a triangle are concurrent. The point of concurrency of the medians.

15. What is the Centroid Conjecture? Example Define Center of Gravity What is the Center of Gravity Conjecture? The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the opposite side. The balanced point of an object. The centroid of a triangle is the center of gravity of the triangular region.

16.  1 st : Find the midpoint of each side  2 nd : Draw a segment from each midpoint to the vertex of the opposite angle.  All three medians intersect at one point.  Point of Concurrency is called Middle Centroid  center of gravity FYI: The distance from vertex to centroid is 2/3 of median The distance from midpoint to centroid is 1/3 of median

17.  1 st : Divide each angle in half  All three angle bisectors intersect at one point.  Point of Concurrency is called Splits angle Incenter

18. Complete practice problems

19. 1. ~Construct each triangle with 1 segment. Median Angle Bisector A B C E D AD = angle bisector CB = median 2. If < EAC = 60, find < EAD. 3. If BA = 10, find AE. 4. If CB = 60, find CJ & JB. J Materials: Ruler Protractor Compass

20. Check Practice Median and angle biscetor Construct medians Form Centroid Center of Gravity

21.  1 st : Find the midpoint of each side  2 nd : Draw a segment from each midpoint that is perpendicular to that side.  All three perpendicular bisectors intersect at one point.  Point of Concurrency is called Forms right angle/splits in half Circumcenter

22. 1st: Draw a segment from vertex that is perpendicular to the opposite side All three altitudes intersect at one point. Point of Concurrency is called Height Orthocenter

23. Perpendicular Bisector Angle Bisector Median Atlitudes

24. Identify each segment and point of concurrency. 1. 3. 2. 4. Angle Bisector Altitude Perpendicular Bisector Median

25. Median Angle Bisector Perpendicular Bisector Altitude

26. Prove and apply theorems and properties of perpendicular bisectors of a triangle Prove and apply theorems and properties of angle bisectors of a triangle. Apply properties of medians of a triangle. Apply properties of altitudes of a triangle.

27. equidistant locus concurrent point of concurrency perpendicular bisector of a triangle circumcenter of a triangle circumscribed angle bisector of a triangle incenter of a triangle inscribed median of a triangle centroid of a triangle altitude of a triangle orthocenter of a triangle

28. When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points.

30. 1. MN  2. BC  3. TU  Given that DE = 20.8, DG = 36.4, and EG =36.4, find EF.

32. 1. BC  2. m<MLK  3. m<EFH given that m<EFG = 50 degrees  Given that YW bisects  XYZ and WZ = 3.05, find WX. 

33. When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle.

34. The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

35. The circumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon.

36. 1. GC  2.a. GM  b. GK  c. JZ 

37. A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle. Materials: Ruler Protractor Compass

38. Unlike the circumcenter, the incenter is always inside the triangle

39. The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point

40. 1. MP and LP are angle bisectors of ∆LMN A. Find the distance from P to MN. B. Find m  PMN. 2. QX and RX are angle bisectors of ΔPQR. Find the distance from X to PQ.

41. 3. A city plans to build a firefighters’ monument in the park between three streets. Draw a sketch to show where the city should place the monument so that it is the same distance from all three streets. Justify your sketch.

42. A city plans to build a firefighters’ monument in the park between three streets. Draw a sketch to show where the city should place the monument so that it is the same distance from all three streets. Justify your sketch.

43. A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Every triangle has three medians, and the medians are concurrent.

44. The point of concurrency of the medians of a triangle is the centroid of the triangle. The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance.

45. 1. In ∆LMN, RL = 21 and SQ =4. A. Find LS B. Find NQ. 2. In ∆JKL, ZW = 7, and LX = 8.1. Find LZ.

46. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle.

47. In ΔQRS, altitude QY is inside the triangle, but RX and SZ are not. Notice that the lines containing the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle.