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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 8. _____ Complete Spiral Review Packet 2 3 Date: _____________________________ 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 uur Example #1: RI bisects ∠SRA . Find the value of x and m∠IRA . suur Example #2: QE is the perpendicular bisector of MU . Find the value of m and the length of ME . uuur Example #3: EA bisects ∠DEV . Find the value of x if m∠DEV = 52 and m∠AEV = 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 m∠EGH = 9z. 7 8 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. 9 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. 10 11 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 12 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 13 14 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 _____________________________________________________ _________________. 15 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 16 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: 17