Today's Class 04/02/2009

Today we started the class by correcting the homework from page 351.
After we finished correcting orally the homework Mrs. Saatkamp corrected some of them on the board as usual.
One exercise that people had problems was number 30 letter C which was the same answer for 28.
And Mrs. Saatkamp gave us a hint on number 32 to know if the triangles are similar or the same after multiplying, adding, subtracting or dividing the coordinations of the vertices.
If you multiply the vertices by a positive number, the figure will enlarge. If you multiply by a negative number that's what will happen:

As you can see the small triangle ( triangle ABC) was the original one and it was multiplied by negative 3. So what happened was that I multiplied the vertices so the other triangle (triangle EFD) got larger, but as it was multiplied by a negative number it was "reflected" to the other. So triangle ABC is similar to triangle EFD

The same thing happens in the division. But the difference is that when you divide the vertices by a positive number the figure will get smaller. But if you divide by a negative number it will not only get smaller but also "reflect". Imagine the original triangle is the bigger triangle (Triangle EFD) I divided by negative 3 and what happens to the other triangle (triangle ABC) was that in it got smaller and it "reflected to the other side".

When you add or subtract a certain number from the vertices (can be positive or negative) of the triangle it will move to the side only, but will stay the "same" triangle in another place. Imagine that the original triangle is the one at the bottom (Triangle ABC) and by adding 3 to the vertices, the triangle stayed the "same" on another place, which is the other triangle (Triangle EFD).

When you subtract a number from the vertices (can be positive or negative) it will only move to another place and stay the "same" as when you add a umber to it. Imagine that the original triangle at the right is the one on top (triangle ABC) so I subtracted 3 from each vertex and what I got was the other triangle (triangle EFD), the bottom one. So Triangle ABC is similar to triangle EFD.

After that we learned the Postulates and Theorems of the chapter and one thing Mrs. Saatkamp said was that we will not have to prove them this semester.

Postulate 7-1: AA Similarity - If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Using the picture:
If:
angle A is congruent to angle E (two top angles)
angle C is congruent to angle D ( the two at the right)
Then:
Triangle ABC is similar to triangle EFD

Theorem 7-1: SSS Similarity - If the measures of thecorresponding sides of two triangles are proportional, then the triangles are similar.
Using the picture:
If:



Then:
Triangle ABC is similarto triangle EFD

Theorem 7-2: SAS Similarity - If the measures of two sides of a triangle are proportional to the measures of two corresponding siides of another triangle and the included angles are congruent, then the triangles are similar.
Using the picture:
If:
Angle B is congruent to Angle F
and
Then:
Triangle ABC is similar to Triangle EFD

Theorem 7-3: Similarity of Triangles is:
Reflexive: Triangle ABC is similar to triangle ABC
Symmetric: Triangle ABC is similar to Triangle EFD, the triangle EFD is similar to triangle ABC
Transitive: Triangle ABC is similar to triangle EFD, triangle EFD is similar to triangle GHI, then triangle ABC is similar to triangle GHI.


After learning the new postulate and theorems we did an exercise which we had this figure below and we had to prove that CB = 4


First you should say that Angles ADE and ABC are congruent by corresponding angles postulate having two parallel lines and a transversal we could conclude that. For the same reason Angles AED and ACB are congruent. After that we could conclude already that both triangles are similar by postulate 7-1 and then we could consider CB as being x and do:



Cross multiply and you will find that x = 4 and then CB = 4

After we finished this, Mrs. Saatkamp gave us the Homework which was page 358 number 13 to 18 and 21 to 25.

One last hint: all congruent triangles, circles, and other polygons are similar but not all similar triangles, circles, and other polygons are congruent.