is similar to 
if 
and 
See
Figure 5.
Each of the following conditions guarantees that
is similar to 
Figure 5
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3. Similar Triangles
| By definition,
is similar to
if
and Each of the following conditions guarantees that |
Figure 5 |
(a) Two angles of one triangle are equal respectively to the corresponding
two angles of the other.
(b) Two sides of one triangle are proportional to the corresponding
sides of the other triangle and the included angles are equal.
(c) The three sides of one triangle are proportional to the corresponding sides of the other.
If we are given that
is similar to
then
As stated in (a), (b), and (c), to prove that
is similar to 
it suffices to show that
and 
or to
show that 
or to show that 
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While sailing around the island in Figure 6a you hit a submerged rock and tore a hole in your boat. You are now stuck on the island and wonder how far it is from point A on the island to point B on the mainland so you can decide if you can risk swimming it. You have a compass, a tape measure, and your trusty HP 48G calculator. How do you estimate the distance from A to B? |
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Solution: From A use the compass to measure the bearing to B. Now walk along the beach to some point C making a line in the sand as you go. See Figure 6b. From C walk along the bearing 180o from the bearing you measured from A to B to some point D from which you can still see B. Put a marker at D and walk strait towards B until you get to the line AC and mark the point E. |
Notice that since AB is parallel to CD, 
Also, 
so 
is similar to 
We now have 
Using the tape measure you find that CE = 21 yd, CD = 67
yd, and AE = 113 yd. Now solve the previous proportion for AB
and substitute the measurements, giving 
The sequence 113 ENTER 67 
on the calculator shows the distance to be a bit over 360 yards.
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