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Section 12-4 Math 366 Lecture Notes Section 12.4 – Similar Triangles and Similar Figures Two figures that have the same shape but not necessarily the same size are similar. The ratio of the corresponding side lengths is the scale factor. Similar Triangles ∆ABC is similar to ∆DEF (∆ABC ∼ ∆DEF) iff ∠ A ≅ ∠ D, ∠ B ≅ ∠ E, ∠ C ≅ ∠ F, and AB AC BC = = . DE DF EF B A E D C F Are all equilateral triangles similar? Theorem 12-10 SSS Similarity for Triangles If corresponding sides of two triangles are proportional, then the triangles are similar. Theorem 12-11 SAS Similarity for Triangles Given two triangles, if two sides are proportional and the included angles are congruent, then the triangles are similar. Theorem 12-12 Angle, Angle (AA) Similarity for Triangles If two angles in one triangle are congruent, respectively, to two angles of a second triangle, then the triangles are similar. 1 Section 12-4 In the figure below, given BD=12, solve for x. D x A 8 E C 5 B Properties of Proportion B B x a z b c y m y w A e C d A f g h C Theorem 12-13 If a line parallel to one side of a triangle intersects the other sides, then it divides those sides into proportional segments. Theorem 12-14 If a line divides two sides of a triangle into proportional segments, then the line is parallel to the third side. Theorem 12-15 If parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on any transversal. Construction Separating a Segment into Congruent Parts Separate AB into three congruent parts. A B 2 Section 12-4 1. Draw any ray, AC , such that A, B, and C are noncollinear. 2. Mark off the given number of congruent segments (of any size) on AC . In this case, we use three congruent segments. 3. Connect B to A3, the endpoint of the last of the three congruent segments. 4. Through A2 and A1, construct parallels to BA3 . Note: There is no construction to trisect an angle. Midsegments of Triangles and Quadrilaterals A midsegment is a segment that connects midpoints of two adjacent sides of a triangle or quadrilateral. Theorem 12-16 The Midsegment Theorem The midsegment is parallel to the third side of the triangle and half as long. B D E A C Theorem 12-17 If a line bisects one side of a triangle and is parallel to a second side, then it bisects the third side and therefore is a midsegment. B D C A E 3 Section 12-4 Indirect Measurements Similar triangles have long been used to make indirect measurement (such as determining the height of the Great Pyramid of Egypt) by using ratios involving shadows. B B’ A’ C’ D A 4 C