Triangles Classification and Measurement Guide

1. Warm-up: How to classify and measure triangles? How to identify angle relationships in triangles in order to measure? How to construct triangles? • 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.

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

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

4. How to classify and measure triangles? How to identify angle relationships in triangles in order to measure? How to construct triangles? Constructions:Angles and Triangles • Use ruler and protractor • Use ruler and compass

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

6. Applying Congruent Triangles Chapter Five

7. 5.1- 5.3 Special Segments in Triangles • To identify and use medians, altitudes, angle bisectors, and perpendicular bisectorsin a triangle.

8. Special Segments • 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. Special Segments • 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. Special Segments • 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. Special Segments • 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. Special Segments • 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. Special Segments • 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. Special Segments • 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. Special Segments • 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. Median Middle • 1st: Find the midpoint of each side • 2nd: 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 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. Angle Bisector Splits angle • 1st: Divide each angle in half • All three angle bisectors intersect at one point. • Point of Concurrency is called Incenter

18. Continue: Median and Angle Bisectors • Complete practice problems

19. Materials: Ruler Protractor Compass Warm Up J • 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.

20. 5.1 - 5.3Special Segments of Triangles & Points of Concurrency Check Practice Center of Gravity • Median and angle biscetor • Construct medians • Form Centroid • Center of Gravity

21. Perpendicular Bisector Forms right angle/splits in half • 1st: Find the midpoint of each side • 2nd: 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 Circumcenter

22. Altitude Height • 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 Orthocenter

23. 5.1 - 5.3Special Segments of Triangles & Points of Concurrency Perpendicular Bisector Angle Bisector Median Atlitudes

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

25. Draw each segment. • Median Angle Bisector • Perpendicular Bisector Altitude

26. Objectives • 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. Vocabulary • 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. Special Segments of Triangles • When a point is the same distance from two or more objects, the point is said to be equidistantfrom the objects. Triangle congruence theorems can be used to prove theorems about equidistant points.

29. Special Segments of Triangles

30. Special Segments of TrianglesFind each measure. • 1. MN  • 2. BC  • 3. TU  • Given that DE = 20.8, DG = 36.4, and EG =36.4, find EF.

31. Special Segments of Triangles

32. Special Segments of TrianglesFind each measure. • 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. Special Segments of Triangles • 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. Special Segments of Triangles • The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

35. Special Segments of Triangles • 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. Special Segments of TrianglesFind each measure. • 1. GC  2.a. GM • b. GK  • c. JZ 

37. Materials: Ruler Protractor Compass Special Segments of Triangles • 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.

38. Special Segments of Triangles • Unlike the circumcenter, the incenter is always inside the triangle

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

40. Special Segments of TrianglesFind each measure. • 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. Special Segments of TrianglesFind each measure. • 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. Special Segments of TrianglesFind each measure. • 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. Special Segments of Triangles • A median of a triangleis 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. Special Segments of Triangles • 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. Special Segments of TrianglesFind each measure. • 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. Special Segments of Triangles • 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. Special Segments of Triangles • 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.