Similar Triangles and Triginimetric Functions Use the figure below. Explain why /_ABC. /_ADE, and /_AFG are similar triangles.

Similar Triangles and Triginimetric Functions Use the figure below. Explain why /_ABC. /_ADE, and /_AFG are similar triangles.

Question
Similarity
asked 2021-02-06
Similar Triangles and Triginimetric Functions Use the figure below.
Explain why \(\displaystyle\triangle{A}{B}{C}.\triangle{A}{D}{E},{\quad\text{and}\quad}\triangle{A}{F}{G}\) are similar triangles.

Answers (1)

2021-02-07
Proof:
Consider the given triangle,
image
The \(\displaystyle\angle{A}\) is part of all three triangles \(\displaystyle\triangle{A}{B}{C},\triangle{A}{D}{E}{\quad\text{and}\quad}\triangle{A}{F}{G}\) where, \(\displaystyle\angle{C}=\angle{E}=\angle{G}={90}^{{\circ}}\).
Since , sum of all angles of a triangle is \(\displaystyle{180}^{{\circ}}\) so from the triangle ABC,
\(\displaystyle\angle{A}+\angle{B}+\angle{C}={180}^{{\circ}}\)
\(\displaystyle\angle{B}={180}^{{\circ}}-\angle{A}-\angle{C}\)
\(\displaystyle={180}^{{\circ}}-\angle{A}-{90}^{{\circ}}\)
\(\displaystyle={90}^{{\circ}}-\angle{A}\)
In the triangle \(\displaystyle\triangle{A}{D}{E}\),
\(\displaystyle\angle{A}+\angle{D}+\angle{E}={180}^{{\circ}}\)
\(\displaystyle\angle{D}={180}^{{\circ}}-\angle{A}-\angle{E}\)
\(\displaystyle={180}^{{\circ}}-\angle{A}-{90}^{{\circ}}\)
\(\displaystyle={90}^{{\circ}}-\angle{A}\)
Similarly, from triangle AFG the angle \(\displaystyle\angle{F}={90}^{{\circ}}—{Z}{A}\).
Therefore, \(\displaystyle\angle{B}=\angle{D}=\angle{F}\).
Since two of the angles of the triangles are equal, the third angle is also equal.
Therefore, the triangles are similar due to AAA similarity criterion.
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