Notes for Homework #3


The definition of similar triangles follows the same format as the definition of congruent triangles. One first defines what it means for a correspondence between the vertices of the triangles to be a `similarity' and then defines two triangles to be similar if there is a similarity between their vertices.


Definition: Let $\Delta ABC$ and $\Delta DEF$ be two triangles. The correspondence MATH is a similarity provided that corresponding angles are congruent and
MATH
In the event the correspondence MATH is a similarity, we write MATH and say that the triangles $\Delta ABC$ and $\Delta DEF$ are similar.

Recall that when establishing two triangles are congruent one does not need to establish that every pair of corresponding angles are congruent and every pair of corresponding sides are congruent. The $ASA,$ $SAS,$ $AAS$ and $SSS$ criteria for congruence tells us that we just need to test some of the pairs in order to conclude that the two triangles are congruent. The situation is similar for similar triangles. Just as in the case for the congruence of triangles, it turns out that in order show that $\Delta ABC$ is similar to $\Delta DEF,$ one does not need to establish that every pair of corresponding angles are congruent and that every pair of corresponding sides are proportional. In particular, one can establish the following:

Theorem: Let $\Delta ABC$ and $\Delta ABC$ be two triangles. Then:

1. AAA criteria: If the corresponding angles are congruent, then $\Delta ABC$ is similar to $\Delta DEF.$

2. SSS criteria: If MATH then $\Delta ABC$ is similar to $\Delta DEF.$

3.SAS criteria: If $\angle A$ is congruent to $\angle D $ and MATH then $\Delta ABC$ is similar to $\Delta DEF.$




During the lecture we also gave a definition of the sine and cosine of an angle. Just for completeness sake, it is included here. It may be useful in problem 2.


Definition: Given $\angle A$, define MATH and MATH as follows:

1. If $\angle A$ is acute, form a right triangle $\Delta ABC$ with right angle at $C$. Then
MATH

2. If $\angle A$ is a right angle, let MATH and cos MATH

3. If $\angle A$ is obtuse and $\angle A^{\prime }$ is a supplement of $\angle A$, then let MATH and cos MATH

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Math 330B - Homework 3

Problem 1: Using the SAS criteria for similarity (see the first page), prove the `midsegment theorem': Let $\Delta ABC$ be a triangle and suppose that $D$ is the midpoint of $\overline{AB}$ and $E$ is the midpoint of $\overline{BC}.$ Prove that MATH is parallel to MATH and $DE=\frac{1}{2}AC.$ (8 pts.)

$\medskip $Problem 2 :Prove the Law of Sines: If $\Delta ABC$ is any triangle, then
MATH

In order to prove this, let $D$ be the foot of the perpendicular dropped from $A$ to MATH. The key to the proof is to use compute $AD$ using MATH and $AC$, compute $AD$ using MATH and $AB$, set the results equal to one another, and then simplify. Notice that there are a number of cases that need to be considered: $B-C-D,$ $C-D-B,$ $D=B,$ $D=C$ and $D-B-C$.

a. Prove the result when $C-D-B.$ Include a diagram that illustrates this case. (4 pts.)

b. Prove the result when $C-B-D.$ Include a diagram that illustrates this case.(4 pts.)

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