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Segment cut by a transversal

Segment cut by a transversal. corollary. Definition . In mathematics , a corollary is a statement which follows readily from a previously proven statement, typically a mathematical theorem . . Segment cut by a transversal.

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Segment cut by a transversal

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  1. Segment cut by a transversal corollary

  2. Definition • In mathematics, a corollary is a statement which follows readily from a previously proven statement, typically a mathematical theorem.

  3. Segment cut by a transversal • If three parallel lines intersect two transversals, then they divide the transversals proportionally.

  4. If three parallel lines intersect two transversals, then they divide the transversals proportionally • Restatement of the corollary • If AD //EF//BC and intersect two transversals (line AB and line DC) , then • DC : DF = AB : AE • BE : EA = CF : FD • AE : DF = BE : CF A D E F B C

  5. Illustrative examples Find the value of x. • Solution: By applying the corollary • 10 : x = 12 : x + 2 Using the principle of proportion • 10(x +2) = x ( 12 ) • 10x + 20 = 12x • 20 = 12x – 10x • 20 = 2x • X = 10 10 12 x X + 2

  6. Another solution By applying the theorem x : x + 2 = 10 : 12 or x : x + 2 = 5 : 6 Using the principle of proportion • 6(x) = 5 (x + 2 ) • 6x = 5x + 10 • 6x – 5x = = 10 • X = 10 10 12 x X + 2

  7. Segment cut by angle bisector Theorem

  8. Segment cut by angle bisector • If a ray bisects an angle of a triangle, it divides the opposite side into segments proportional to the other two sides.

  9. Exploration • Construct any triangle. Construct an angle bisector in the triangle and draw the segment along the angle bisector from the vertex to the intersection with the opposite side. • Measure the ratio of the adjacent sides . • Measure the ratio of the segments cut off by the bisector on the opposite side. • Repeat for many triangles .

  10. ILLUSTRATION • The bisector of an angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides.

  11. ILLUSTRATION C • for any triangle ABC, the bisector of the angle at C divides the opposite side into segments of length x and y such that A D B

  12. ILLUSTRATION C or x : y = a : b x a y b A D B

  13. If a ray bisects an angle of a triangle, it divides the opposite side into segments proportional to the other two sides. A • Restatement of the theorem • If AD bisects angle BAC of triangle ABC, then … • BD : DC = AB : AC • BD : BC = AB : AB + AC • DC : BC = CA : CA + AB B C D

  14. Illustrative examples Find the value of x if a =10, b = 15 and y = 12.

  15. solution By applying the theorem x : y = a : b or x : 12 = 10 : 15 Using the principle of proportion • 15(x) = 12 (10 ) • 15x = 120 • x = 120 15 x = 8

  16. another solution By applying the theorem x : a = y : b or x : 10 = 12 : 15 Using the principle of proportion • 15(x) = 12 (10 ) • 15x = 120 • x = 120 15 x = 8

  17. another solution By applying the theorem x : x + y = a : a +b or x : x+12= 10 : 10 +15 x : x+12= 10 : 25 x : x+12= 2: 5 Using the principle of proportion • 5(x) = 2 (x +12 ) • 5x = 2x + 24 • 5x – 2x = 24 3x = 24 x = 8

  18. QUIZ 1 FIND THE VALUE OF a. 1. If x = 6 , y = 14 and b = 20. X + 1

  19. QUIZ 2 FIND THE VALUE OF x. X + 1 X + 4 13 19

  20. Assignment • Test yourself nos. 1- 5, page 159. • Geometry textbook • ( one- fourth )

  21. Assignment • Find the value of y. y 24 12 20

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