Geometry 7 6 Proportional Lengths Proportional Lengths n

Proportional Lengths n AC and XZ are divided proportionally if… . . . 2

Triangle Proportionality Thm. n big A small A whole A If a line parallel

Corollary If three // lines intersect two transversals, … then they divide the transversals

Triangle Angle Bisector Thm. n If a ray bisects an angle of a triangle,

Directions: Use the given information and the triangle above to find the missing segment.

Directions: Use the given information and the diagram to find x. 2. AB =

Directions: 2. RP = x is and angle bisector of PQ = 12. 5

HW n P. 271 (1 -7) P. 272 -273 (1 -14, 20, 21) Quiz

An Example 16 4 12 24 18 15 20 15 5 5 20 15

Solve for x. (figure not to scale) 15 Reduce by 3 27 15 12

Reduce by 9. Solve for x. 18 27 20 x 30 18 27 x

Solve for x. Reduce by 3. 12 15 x = 18 - x 4

Slides: 14

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Geometry 7. 6 Proportional Lengths

Proportional Lengths n AC and XZ are divided proportionally if… . . . 2 A BC = XY YZ 9 B C 4 X AB Y 18 . Z Example: 2 9 = 4 18

Triangle Proportionality Thm. n big A small A whole A If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. = big B whole A small B = big B big A whole B side C 1 small A small B whole B side C 2 big A whole A small B = All of these proportions, and their inverses, work. The key is to use the easiest one to solve each problem. whole B big A big B = whole A whole B big A = big B = side C 1 whole A whole B side C 2 Think of it as two separate similar triangles.

Corollary If three // lines intersect two transversals, … then they divide the transversals proportionally. n a b c d a b = c d

Triangle Angle Bisector Thm. n If a ray bisects an angle of a triangle, … then it divides the opposite side into segments proportional to the other two sides. a c b d a b = c d

Directions: Use the given information and the triangle above to find the missing segment. D 3. GE = 15 DE = 27 HF = 20 DH = ? G E 4. DE = 20 HF = 4 DH = 12 DG = ? H F

Directions: Use the given information and the diagram to find x. 2. AB = 27 BC = 18 DE = x + 10 A EF = x B C 3. AB = 25 – x BC = x DE = 16 EF = 4 D E F

Directions: 2. RP = x is and angle bisector of PQ = 12. 5 RS = 8 P . Find x. SQ = 10 R 3. RP = 10 PQ = 20 RS = x RQ = 15 S Q

HW n P. 271 (1 -7) P. 272 -273 (1 -14, 20, 21) Quiz 7. 4 -7. 6 Tomorrow

A few from the HW n P. 272 #5, #14

An Example 16 4 12 24 18 15 20 15 5 5 20 15 20 = 12 = 4 = 5 4 16 5 4 = 15 = 12 12 = 20 16 12 16 15 20 16 = 18 24

Solve for x. (figure not to scale) 15 Reduce by 3 27 15 12 = 20 20 12 x x 5 times 4 equals 20 4 times 4 equals = x x = 16

Reduce by 9. Solve for x. 18 27 20 x 30 18 27 x = x + 10 2 3 x = x + 10 2(x + 10) = 3 x 2 x + 20 = 3 x x = 20

Solve for x. Reduce by 3. 12 15 x = 18 - x 4 5 x = 18 - x 12 15 x 8 18 - x 10 18 4(18 – x) = 5 x 72 – 4 x = 5 x 9 x = 72 x=8