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| 3. Similar Triangles
By definition, Each of the following conditions guarantees that (a) Two angles of one triangle are equal respectively to the corresponding two angles of the other. |
 Figure 6 |
is similar to 
if
and 
See Figure 6.
is similar to 
(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 
Calculator Example 1.3.1
Figure 7a |
While sailing around the island in Figure 7a 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 49G+ calculator. How do you estimate the distance from A to B? 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 7b. 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,
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Figure 7b |
(See 
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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