1. The four triangles are ∆ACB, , ∆DGB ∆DFE, and ∆BHE .
2. In order for triangles to be similar, certain relationships must exists between two
or three pairs of corresponding parts.
3. a. ∆ ∆ ACB DGB ∆DFE BHE ∆ ;Possible plan: Use the parallel lines and
transversals to show that ∠A ≅ ∠EDF ≅ ∠EBH and ∠E ≅ ∠ABC.Then use the
AA Similarity Postulate.
b. AB DE DB BE
AC DF DG BH = = =
4. CZ , ZY , and YX are proportional as stated in the Corollary to the Side-Splitter
Theorem: If three parallel lines intersect two transversals, then the segments
intercepted on the transversals are proportional.
5. Possible plan: Use the Similar Triangles and the Segment Addition Postulate to
write and solve a proportion to find the length ofGZ .
6. a. BE = 15 in., BX = 9 in., EX = 12 in.
Possible Plan: Use the fact that the diagonals of a kite are perpendicular to
show that ∆BEC is a right triangle with right∠BEC. Then use the fact
that EX is the altitude to the hypotenuse to write and solve proportions to find
the lengths of BX and EX.
b–d. XEFY: XE = 12 in., EF = 7.5 in., FY = 7.5in., XY = 6 in.,
YFGZ: YF = 7.5 in., FG = 2.5 in., GZ = 6 in., YZ = 2 in.
∆ZGC :ZG = 6 in., GC = 10 in., ZC = 8 in.
Possible plan: Use the fact that ∆ ∆ CXE EXB to write and solve a
proportion to find the length ofCE. Then use the extended ratio to write and
solve an equation to find the lengths of XY,YZ, and ZC . From the Corollary
to the Side-Splitter Theorem, EF: FG: GC is also 3:1:4. Use the extended
ratio to write and solve an equation to find the lengths of EF, FG, and GC .
By the AA similarity Postulate, ∆CXE ∆CYF CZG ∆ . Use the Similar
Triangles and the Segment Addition Postulates to write and solve proportions
to find the lengths ofYF and ZG