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# 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.

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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, 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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