Triangle Similarity
We will use the SAS theorem to prove that Triangle TSG and Triangle PHG are similar.
First, we need to define the SAS similarity theorem:
<em>The Side-Angle-Side (SAS) Theorem states if two sides of one triangle are proportional to two corresponding sides of another triangle, and their corresponding included angles are congruent, the two triangles are similar.</em>
To use this theorem, we need to prove:
* Two sides of the triangles are in the same proportion
* The included angle between them is congruent with the other corresponding angle.
In triangle TSG, the longest side has a length of 7.5 + 5 = 12.5 feet
In triangle PHG, the longest (and therefore corresponding) side has a length of 5 feet.
The ratio between them is 12.5 / 5 = 2.5
In triangle TSG, the horizontal length is 4.5 + 3 = 7.5 feet
In triangle PHG, the horizontal (and therefore corresponding) side has a length of 3 feet.
The ratio between them is 7.5 / 3 = 2.5
Since the ratio measured on both corresponding sides is the same, we have satisfied the first condition of the SAS theorem.
The included angle is G, and it's the same for both triangles.
The reflexive property guarantees both triangles have congruent corresponding angles:
Angle G is congruent with (itself) Angle G
Thus, we have met the conditions for the SAS similarity theorem and therefore proven Triangle TSG and Triangle PHG are similar.
