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Geometry Lesson Notes 6.4A ____________________ Objective: Use proportional parts of triangles to solve problems. Triangle Proportionality Theorem: If a line is parallel to one side of a triangle, then it separates the two other sides into segments of proportional lengths. Note: This proportion is not the same as the scale factor for the s. A Practice: Which triangles are similar? Why? What is the ratio of their sides? 30 20 What is the ratio of the segments? B D 28 10 15 C E 42 Example 1 (p 308): Find the length of a Side F In DFG, EH || FG , FE = 9, ED = 21, and HG = 6. Find DH. E D H G Note: This can, of course, be solved using corresponding sides of similar triangles. 478158526 Page 1 of 6 Converse of the Triangle Proportionality Theorem: If a line separates two sides of a triangle into corresponding segments of proportional lengths, then the line is parallel to the third side of the triangle. Example 2 (p 308): Determine if Lines are Parallel In HKM, HM = 15, HN = 10, and HJ is twice the length of JK . Is MK || NJ ? Explain. M K N J H Midsegment of a triangle: a segment with its endpoints at the midpoints of two sides of a triangle. Theorem 6.6 Triangle Midsegment Theorem: A midsegment of a triangle is parallel to the third side of the triangle and is half the length of the third side. Practice: x+6 3x − 8 478158526 Page 2 of 6 Example 3 (p 309): Midsegment of a Triangle on the Coordinate Plane Triangle ABC has vertices A(–4, 1), B(8, –1), and C(–2, 9). DE is a midsegment of ABC. a. Find the coordinates of D and E. b. Verify that AC is parallel to DE . c. Verify that DE is ½AC. CW pp 311-312 #5-10 HW 478158526 A4a pp 312-313 #14-25 Page 3 of 6 Geometry Objective: Lesson Plan 6.4B ____________________ Divide a segment into congruent parts. Corollary 6.1: If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. AB XY AC XZ A AB XY BC YZ BC YZ AC XZ X B Y C Z Notice also, that the properties of proportions allow us tomake many more true proportions. IMPORTANT: You can not use the parallel segments in these ratios!!! Practice: What values could x and y have? x 6 10 478158526 y Page 4 of 6 Example 4 (p310): Proportional Segments The distance from A Street to E Street along X Avenue is 3800 ft. The distance between the same two streets is 4430 ft. along Y Avenue. The distance from C Street to D Street is 411 ft. along X Avenue. What is the distance between the two streets along Y Avenue? A Street B Street C Street D Street 478158526 YAvenue XAvenue E Street Page 5 of 6 E Corollary 6.2: If three or more parallel lines cut off congruent segments on one transversal, then they cut of congruent segments on every transversal. Practice: 3x – 8 x+y 2x + 5 CW p 312 #11-13, 33-34 HW A4b fms-Geometry Worksheet 6.4 A4c Lesson 6-4 Skills Practice / Practice Prepare for Quiz 6.3-6.4 478158526 Page 6 of 6