Geometry. Unit 5 Relationships in Triangles. Name:

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1 Geometry Unit 5 Relationships in Triangles Name: 1

2 Geometry Chapter 5 Relationships in Triangles ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** 1. (5-1) Bisectors, Medians, and Altitudes Page all 2. (5-1) Bisectors, Medians, and Altitudes Pages all 3. (5-1) Bisectors, Medians, and Altitudes 5-1 Practice Worksheet 4. (5-2) Inequalities and Triangles Pages , 29-34, 37-43, 46, (5-2) Inequalities and Triangles 5-2 Practice Worksheet 6. (5-4) The Triangle Inequality Pages even, 57, Chapter 5 Review WS 2

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4 Section 5 1: Bisectors, Medians, and Altitudes Notes Part A Perpendicular Lines: Bisect: Perpendicular Bisector: a line, segment, or ray that passes through the of a side of a and is perpendicular to that side Points on Perpendicular Bisectors Theorem 5.1: Any point on the perpendicular bisector of a segment is from the endpoints of the. Example: Concurrent Lines: or more lines that intersect at a common Point of Concurrency: the point of of concurrent lines Circumcenter: bisectors of a triangle the point of concurrency of the 4

5 Circumcenter Theorem: the circumcenter of a triangle is equidistant from the of the triangle Example: Points on Angle Bisectors Theorem 5.4: Any point on the angle bisector is from the sides of the angle. Theorem 5.5: Any point equidistant from the sides of an angle lies on the bisector. Incenter: the point of concurrency of the angle of a triangle Incenter Theorem: the incenter of a triangle is from each side of the triangle Example: 5

6 Example #1: RI bisects SRA. Find the value of x and m IRA. Example #2: QE is the perpendicular bisector of MU. Find the value of m and the length of ME. Example #3: EA bisects m AEV = 6x 10. DEV. Find the value of x if m DEV = 52 and 6

7 Example #4: Find x and EF if BD is an angle bisector. Example #5: In DEF, GI is a perpendicular bisector. a.) Find x if EH = 19 and FH = 6x 5. b.) Find y if EG = 3y 2 and FG = 5y 17. c.) Find z if m EGH = 9z. 7

8 CRITICAL THINKING 1.) Draw a triangle in which the circumcenter lies outside the triangle. 2.) For what kinds of triangle(s) can the perpendicular bisector of a side also be an angle bisector of the angle opposite the side? 3.) For what kind of triangle do the perpendicular bisectors intersect in a point outside the triangle? 8

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10 Date: Section 5 1: Bisectors, Medians, and Altitudes Notes Part B Median: a segment whose endpoints are a of a triangle and the of the side opposite the vertex Centroid: the point of concurrency for the of a triangle Centroid Theorem: The centroid of a triangle is located of the distance from a to the of the side opposite the vertex on a median. Example: Example #1: Points S, T, and U are the midpoints of DE, EF, and DF, respectively. Find x. 10

11 Altitude: a segment from a to the line containing the opposite side and to the line containing that side Orthocenter: the intersection point of the Example #2: Find x and RT if SU is a median of RST. Is SU also an altitude of RST? Explain. Example #3: Find x and IJ if HK is an altitude of HIJ. 11

12 CRITICAL THINKING 1.) R(3, 3), S(-1, 6), and T(1, 8) are the vertices of RST, and RX is a median. a.) What are the coordinates of X? b.) Find RX. c.) Determine the slope of RX. d.) Is RX an altitude of RST? Explain. 2.) Draw any XYZ with median XN and altitude XO. Recall that the area of a triangle is one-half the product of the measures of the base and the altitude. What conclusion can you make about the relationship between the areas of XYN and XZN? 12

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14 Date: Section 5 2: Inequalities and Triangles Notes Definition of Inequality: For any real numbers a and b, if and only if there is a positive number c such that. Example: Exterior Angle Inequality Theorem: If an angle is an angle of a triangle, then its measures is than the measure of either of its remote interior angles. Example: Example #1: Determine which angle has the greatest measure. Example #2: 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 8 b.) all angles whose measures are greater than m 2 14

15 Theorem 5.9: If one side of a triangle is than another side, then the angle opposite the longer side has a measure than the angle opposite the shorter side. Example #3: angles. Determine the relationship between the measures of the given a.) RSU, SUR b.) TSV, STV c.) RSV, RUV Theorem 5.10: If one angle of a triangle has a measure than another angle, then the side opposite the greater angle is than the side opposite the lesser angle. Example #4: Determine the relationship between the lengths of the given sides. a.) AE, EB b.) CE, CD c.) BC, EC 15

16 CRITICAL THINKING 1.) Find The Error: Hector and Grace each labeled QRS. Who is correct? Explain. 2.) Write and solve an inequality for x. 16

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18 Name Chapter 5 (5.4) Period Use your paper strips to determine whether a triangle can be formed. Complete the following chart using the correct values. Orange = 2 inches Yellow = 3 inches Blue = 4 inches Green = 5 inches Side measure First side Second side Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 Trial 7 Third side Is it a triangle? What can you conclude from the data in the table above? Complete the following sentence: In order to have a triangle, the sum of two smallest sides must be. 18

19 Date: Section 5 4: The Triangle Inequality Notes Triangle Inequality Theorem: The sum of the lengths of any two sides of a is than the length of the third side. Example: Example #1: Determine whether the given measures can be the lengths of the sides of a triangle. a.) 2, 4, 5 b.) 6, 8, 14 Example #2: Find the range for the measure of the third side of a triangle given the measures of two sides. a.) 7 and 9 b.) 32 and 61 19

20 Theorem 5.12: The perpendicular segment from a to a line is the segment from the point to the line. Example: Corollary 5.1: The perpendicular segment from a point to a plane is the segment from the point to the plane. Example: 20

21 CRITICAL THINKING 1.) Find The Error: Jameson and Anoki drew EFG with FG = 13 and EF = 5. They each chose a possible measure for GE. Who is correct? Explain. 2.) Find three numbers that can be the lengths of the sides of a triangle and three numbers that cannot be the lengths of the sides of a triangle. Justify your reasoning, and include a picture. 21