Geometry. Relationships in Triangles. Unit 5. 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 8. Complete Spiral Review Packet 2
3 3
4 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. 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
5 Circumcenter Theorem: the circumcenter of a triangle is equidistant from the of the triangle 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 5
6 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 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 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 #1: Points S, T, and U are the midpoints of DE, EF, and DF, respectively. Find x. 9
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. 10
11 11
12 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. 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 #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
13 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 13
14 14
15 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
16 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 #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
17 Theorem 5.12: The perpendicular segment from a to a line is the segment from the point to the line. Corollary 5.1: The perpendicular segment from a point to a plane is the segment from the point to the plane. 17
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1. Given: a b, b c a c Guidance: Draw a line which intersects with all three lines. 2. Given: a b, c a a. c b b. Given: d b d c 3. Given: a c, b d a. α = β b. Given: e and f bisect angles α and β respectively.
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