Geometry: An Interactive Journey to Mastery



8. D௘ͽ L /DEHOWKHGLDJUDPDVVKRZQLQFigure S.9.4.
   ii. ++PBQ~ ABCEHFDXVHRI6$6ͼ௘x and 3x, y and 3y,
   and share angle B௘ͽ
   iii. a = b because they are matching angles in ~ triangles.
   iv. PQ AC& because the same-side interior angles
   marked bDQGíaVXPWRƒ
   E௘ͽ 7KHVFDOHIDFWRUEHWZHHQWKHWZRWULDQJOHVLVk = 3, so
   AC ˜3.PQ


9. D௘ͽ %RWKWULDQJOHVKDYHDƒDQJOHDQGVKDUHDQJOHC. By AA, they are
   VLPLODU௘6HHFigure S.9.5௘
   E௘ͽ 6LGHx in the big triangle matches side y in the small triangle.
    6LGHy + z in the big triangle matches side x in the small triangle.
   Because matching sides in ~ triangles come in the same ratio,
   xyx yz.
   Algebra gives x = yͼ௘y + z௘ͽ


10. L 'UDZDGLDJRQDODQGODEHOWKHDQJOHVDVVKRZQLQ
    Figure S.9.6.
    ii. a = b because they are alternate interior angles for top
    and bottom parallel lines.
    iii. c = d because they are alternate interior angles for left
    and right parallel lines.
    iv. The two triangles are ~ because AA.
    v. k = 1 because they share the diagonal.
    vi. x = y because k = 1.

A

a

b

x y

P

B

Q

C

ía

3 x

3 y

Figure S.9.4

A

BC

D

x

y

z

Figure S.9.5

x a

c

db y

Figure S.9.6