Proportional Parts Advanced Geometry Similarity Lesson 4

If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths. Triangle Proportionality Theorem If,

endpoints are the midpoints of two sides Midsegment

Triangle Midsegment Theorem A midsegment of a triangle is parallel to one side of the triangle, and its length is one-half the length of that side. and

Example: Find x, BD, and AE.

Proportional Segments If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

Example: Find x.

If two triangles are similar, then their perimeters are proportional to the measures of the corresponding sides. Proportional Perimeters

EXAMPLE: If ∆DEF ∼ ∆GFH, find the perimeter of ∆DEF.

If two triangles are similar, then the measures of the corresponding altitudes, angle bisectors, and medians are proportional to the measures of the corresponding sides. Special Segments of Similar Triangles

EXAMPLE: In the figure, ∆EFD ~ ∆JKI. is a median of ∆EDF and is a median of ∆JIK. Find JL if EF = 36, EG = 18, and JK = 56.

EXAMPLE: The drawing below illustrates two poles supported by wires. ∆ABC ~ ∆GED. and Find the height of pole

An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides. Angle Bisectors segments with endpoint A segments with endpoint C

EXAMPLE: Find x if AB = 10, AD = 6, DC = x, and BC = x + 6.