BASIC PROPORTIONALITY THEOREM • If a line intersects two sides of a triangle and is parallel to the third side, then it divides the first two sides proportionally.

RESTATEMENT OF THE THEOREM B • If a line (EF) intersects two sides ( AB & CB) of a triangle (ABC) and is parallel to the third side( AC ), then it divides the first two sides proportionally. • Thus, E F C A

Converse of the Triangle Proportionality Theorem B • If a line divides two sides of a triangle proportionally, then it is parallel to the third side. E F C A

Parallel Lines Proportionality Theorem • If three parallel lines intersect two transversals, then they divide the transversals proportionally.

VERIFYING A PROPORTIONS( an example) B 1. BE : EA = BF : FC • 15 : 5 = 12 : 4 • By simplifying, • 3 : 1 = 3 : 1 15 12 6 E F 4 5 8 C A

VERIFYING A PROPORTIONS B 2. BE : BA = BF : BC • 15 : 20 = 12 : 16 • By simplifying, • 3 : 4 = 3 : 4 15 12 6 E F 4 5 8 C A

VERIFYING A PROPORTIONS B 3. BA : EA = BC : FC • 20 : 5 = 16 : 4 • By simplifying, • 4 : 1 = 4 : 1 15 12 6 E F 4 5 8 C A

VERIFYING A PROPORTIONS B 4. BE : BF = EA : FC • 15 : 12 = 5 : 4 • By simplifying, • 5 : 4 = 5 : 4 15 12 6 E F 4 5 8 C A

VERIFYING A PROPORTIONS B 5. FC : EA = BC : BA • 4 : 5 = 16 : 20 • By simplifying, • 4 : 5 = 4 : 5 15 12 6 E F 4 5 8 C A

VERIFYING A PROPORTIONS B 6. EF : AC = BF : BC • 6 : 8 = 12 : 16 • By simplifying, • 3 : 4 = 3 : 4 15 12 6 E F 4 5 8 C A

VERIFYING A PROPORTIONS B 6. EF : AC = BE : BA • 6 : 8 = 15 : 20 • By simplifying, • 3 : 4 = 3 : 4 15 12 6 E F 4 5 8 C A

Exercises • GIVEN: DE // BC, • AD = 9, AE = 12, DE = 10,DB = 18. • Find, • BC, AC and CE. A 9 12 D E 10 18 C B

Solution • Find BC, • BC : DE = BA : DA • BC : 10 = 27 : 9 or • BC : 10 = 3 : 1 • Applying principle of proportion • BC(1) = 10(3) • BC = 30 A 9 12 D E 10 18 30 C B

Solution • Find AC, • AC : AE = BA : DA • AC : 12 = 27 : 9 or • AC : 12 = 3 : 1 • Applying principle of proportion • AC(1) = 12(3) • AC = 36 A 9 12 D E 10 18 C B

Solution • Find CE, • CE : AE = BD : DA • CE : 12 = 18 : 9 or • CE : 12 = 2 : 1 • Applying principle of proportion • CE(1) = 12(2) • CE = 24 A 9 12 D E 10 24 18 C B

Solution • Another way to find CE, • CE = AC – AE • Hence, AC =36, then • CE = 36 – 12 • CE = 24 A 9 12 D E 10 24 18 C B

Quiz • Solve the following problem. Show your solution.( one –half crosswise) • 1. Uncle Tom plans to divide an 80- meter rope into three pieces in the ratio 3 : 5 : 8. what will be the length of each piece?

QUIZ • 2. In the figure, find the values of x and y. 30 y 15 12 x 10