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## Geometry (all content)

### Course: Geometry (all content)>Unit 12

Lesson 5: Solving problems with similar and congruent triangles

# Using similar & congruent triangles

Sal uses the similarity of triangles and the congruence of others in this multi-step problem to find the area of a polygon. Created by Sal Khan.

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• Since GF + FE = 18, and GC = 24, can I use trigonometric functions to find the length of EC? • how will you find the answer if it does not give you this information? • So... how do we know from the problem that line CG comes down to hit the base of triangle ACE at a right angle? Why can't it be slanted? Given that the fact that it creates a right angle is important for establishing similarity, I think it's a kind of important jump, and I don't see how you can deduce that from the problem. • Say I have a RIGHT ANGLED TRIANGLE. If I cut a PERPENDICULAR LINE with 90* angle at base, will the smaller triangle ALWAYS be similar to the entire triangle? • At around seven minutes, Sal proves that triangle AHB is similar to triangle AGC. I had realized that since triangle AHB was congruent to triangle EFD, then triangle AGC is congruent to triangle EGC through AAS (<A is congruent to <C, <AGC and <EGC are both right angles, and CG is congruent to itself). Why did Sal take the extra step? • I got a dumb question: I know how to do this kind of problem but don't know how to do the practice problems, why is that? Maybe I used the wrong numbers to set up a proportion? • I am wondering, my solution arrived at the same answer though I used a different path, or algorithm, if I remember

If Triangle ACE = Isosceles triangle, then CEG = CAG
they are also similar, as are CEG ~ DFE, given this info
and the additional info of, GF + FE = GE * GC/2 = a of CGE
ge/gc : fe/df = (1) 18/24 :6/x (2) 18x = 144 (3)18x/18 = 144/18 = x = 8 = DF = BH

(&statement)because of the Isosceles triangle rule that CE = CA,
and angle = DEF & angle DFE (AA) as related to the above congruency (if =(&statement) then BA = DE, because of AA within an isosceles triangle

then we plug in all the values and get the same answer as your proof.

if this wasn't an isosceles triangle then your method would be required right?

but can't I assume these things? within the context of an isosceles triangle and two
alternate interior angles that must be congruent or equal triangles -this would seem like the faster way, though I am curious about another example with a scalene triangle?

addendum - I should have submitted the video link and time, will correct soon...

thanx,

j • How do you find corresponding sides in similar triangles? • Corresponding sides? You do cross multiplication. In similar triangles, the fraction of the values of one from each triangle on one particular side is equal to another. Here, if two triangles are similar, and one has a side of 5, and the other has the exact same side (but with a different value) has a value of 8, then the fraction can be put as either 5/8 or 8/5, and the other side you want to find can be put in the same way. I put the value of the side on the triangle with 8 on the top, assuming I do 8/5, then put the other at the bottom. There's going to be an unknown value, so cross multiply. If it ends up as 8/5*6/x, then 8*x, 5*6, so 8x=30. And the answer to that is 15/4. Sorry about the long answer, but hope if helps!
(If I do turn out to be wrong, then I strongly apologize, just in case.)
• I have seen that the triangles have to be symmetric. Can this formula work for any-other triangles?
(1 vote) • Which formula?

Triangles do not have to be symmetric to be similar and congruent. Congruent means that a triangle has the same angle measures and side lengths of another, but it might be positioned differently, maybe rotated.

Similar only means the angles are the same. If you have two triangles that have the same angle measures then they will be similar, the sides will be "scaled" versions of them. This means the sides of the original will be multiplied or divided by some number. So say the side lengths are 3, 4, 5. a scaled up version could be 6, 8, 10 or 15, 20, 25 or 3/2, 2, 5/2 or 3/5, 4/5, 1. Similar triangles just need the same angle measures. 