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Geometry Unit 5 Relationships in Triangles Name:________________________________________________ 1 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 235 1-13 all 2. ____ (5-1) Bisectors, Medians, and Altitudes – Pages 243-244 11-22 all 3. ____ (5-1) Bisectors, Medians, and Altitudes – 5-1 Practice Worksheet 4. ____ (5-2) Inequalities and Triangles – Pages 252-253 17-25, 29-34, 37-43, 46, 47 5. ____ (5-2) Inequalities and Triangles – 5-2 Practice Worksheet 6.____ (5-4) The Triangle Inequality – Pages 264-266 14-36 even, 57, 58 7. _____ Chapter 5 Review WS 2 Date: _____________________________ 3 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: the point of concurrency of the _____________________ bisectors of a triangle 4 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 Example #1: RI bisects SRA . Find the value of x and mIRA . Example #2: QE is the perpendicular bisector of MU . Find the value of m and the length of ME . Example #3: EA bisects DEV . Find the value of x if mDEV = 52 and mAEV = 6x – 10. 6 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 mEGH = 9z. 7 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 9 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 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 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 13 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 m8 b.) all angles whose measures are greater than m2 14 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: Determine the relationship between the measures of the given angles. 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 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 17 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 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 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 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