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Day 50

Day 50. Parts of Similar Triangles. Today’s Agenda. Triangle Proportionality Triangle Midsegment Proportional parts of Parallel Lines Angle Bisector Theorem. Review of Similarity. Two figures are similar if they have the same shape.

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Day 50

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  1. Day 50 Parts of Similar Triangles

  2. Today’s Agenda • Triangle Proportionality • Triangle Midsegment • Proportional parts of Parallel Lines • Angle Bisector Theorem

  3. Review of Similarity • Two figures are similar if they have the same shape. • The corresponding angles of similar figures are congruent. • The corresponding sides of similar figures are in a consistent ratio. • There are 3 ways to tell if triangles are similar: • AA • SSS • SAS

  4. Triangle Proportionality Theorem • If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional lengths. A D C E B

  5. Triangle Proportionality Theorem • Proof: • Because AB ll DE, A  CDE and B  CED. (Corresponding s) • Therefore ABC  DEC (AA) • (Def. of  polygons) • (Seg. Addition) A D C E B

  6. Triangle Proportionality Theorem • Example: • Solve for x. A 3 D 7.5 C x E 2.5 B

  7. Converse of Triangle Proportionality Theorem • The converse of the Triangle Proportionality Theorem is true: • If a line intersects two sides of a triangle and divides the sides into segments of proportional lengths, then that line is parallel to the third side • If then AB ll DE. A D C E B

  8. A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle. The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. DE ║ AB, and DE = ½ AB We have proven the Midsegment Theorem with a coordinate proof, but it’s also a natural result of the previous theorem. Midsegment Theorem

  9. What would happen if you drew all three midsegments? How do the lengths of the midsegments compare to the other lengths? So what have we formed? Four congruent triangles (SSS). How do they relate to the original triangle? They are all similar to the original – each side is scaled by a factor of ½. Connect all midpoints R U V T W S

  10. Proportional Parts of Parallel Lines • If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. • Note: If AC and DF are extended up, they will form a triangle. • If AD ll BE ll CF,then A D B E F C

  11. Proportional Parts of Parallel Lines • As a corollary, if three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. • If AD ll BE ll CF,and AB  BC, thenDE  EF. A D B E F C

  12. Proportional Parts of Parallel Lines • We can use this property to divide a segment into any number of congruent pieces. • Get out your compass and straight edge; follow the video. (Your book uses a slightly different, but equally valid technique on p. 488.) • The idea is to create parallel lines spaced equally apart.

  13. Special Segments of Similar Triangles • If two triangles are similar by a scale of a:b, then the corresponding: • altitudes • medians • angle bisectors • have the same ratio of similarity (i.e., are proportional to the corresponding sides).

  14. Angle Bisector Theorem • If you draw an angle bisector of a triangle, you separate the opposite side into two segments that are proportional to the lengths of the other two sides. • We’ll bisect A. • The Angle BisectorTheorem says: A B C D

  15. Example • Say we know these measurements: • And we want to solve for x. • According to the theorem: • Now solve. A 9 6 B C D 2 x

  16. Another Example • How would we solve this? Z 15 W 20 6 Y a X

  17. Proof • Take any triangle. • Bisect an angle. • We are going to create a segment parallel to MN that passes through point O. • Now we extend the angle bisector. • NMP  OQP (why?) • MNP  QOP (why?) • MOQ is isosceles (why?) • So, setting up a proportion based on the corresponding sides of similar triangles: M b a N c P d O a Q

  18. Homework 28 • Workbook, p. 90-92

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