Triangle Proportionality Theorem Help?

[ferrari]

Hello. I am taking geometry this year. I am not good with math, and I was wondering how to do the Triangle Proportionality Theorem? I tried looking up how to do it on YouTube, , but the videos were not correct, my teachers said that I did stuff wrong. Overall I don't understand what the heck I am doing, and I am hoping someone can explain it to me step by step, as I cannot receive information in blobs, it has to be completely separated out and into individual steps. Also I don't know what numbers go where and how often that applies to a situation of a problem. Can someone please help me?

 

[Harry_the_cat]

Ok. All of these diagrams involve similar triangles.

Firstly a question: Do you know the angle rules regarding parallel lines - particularly corresponding angles?

If not, google "corresponding angles on parallel lines" to familiarise yourself with that rule.

Then let me know and I'll continue.

 

[ferrari]

i know that rule

 

[Harry_the_cat]

i know that rule

Ok good.
Now label your triangles. In each diagram there are two triangles - a big one and a small one. Label the big one ABC and the small one AEF, for example. Use A for the common vertex (point).

Now mark the angles that are the same (using corresponding angles on parallel lines) with the same symbol, a dot, a star or something like that.

Ok so let's look at Q6 your first attachment. A is at the top. Left bottom point is B, right bottom point is C. E is the point between A and B, F is the point between A and C....so far so good.

Now, because three angles of triangle ABC are congruent to three angles of AEF, then those two triangles are similar.
That is Triangle ABC is similar to Triangle AEF. (Note the order in which I have written these - here angle B = angle E, angle C = angle F and, of course angle A = angle A.)
So I've written ABC is similar to AEF (not eg ABC is similar to AFE - where the angles don't match up).

Once you've labelled identified the two similar triangles ABC and AEF, you can use the "triangle proportionality theorem" to find missing information:

If ABC and AEF are similar, then $\displaystyle \frac{AB}{AE} = \frac{AC}{AF}=\frac{BC}{EF}$.

In Q6, you know that AB = 3+5=8, AE=3, BC=8 and you are asked to find x=EF.

So using $\displaystyle \frac{AB}{AE} =\frac{BC}{EF}$....(you don't need the middle fraction in this case)

$\displaystyle \frac{8}{3} =\frac{8}{EF}$

So x = EF = 3.

All the others are much the same:
1. Label the vertices
2. Mark the congruent angles
3. Identify the two similar triangles (labelling them in the correct order.
4. Write down the "triangle proportionality theorem"
5. substitute in the values you know.
6. Work out the one required!

Give it a go!