Proving Similarity in the figure DEFG is a square. Prove the following: /_ADG~/_GCF Given: The given figure is, 12210202691.jpg

Question
Similarity
Proving Similarity in the figure DEFG is a square. Prove the following:
$$\displaystyle\triangle{A}{D}{G}\sim\triangle{G}{C}{F}$$
Given:
The given figure is,

2021-02-03
Approach:
Two triangles are similar if their vertices can be matched up so that corresponding angles are congruent. In this case corresponding sides are proportional.
If the two angles of the triangles are same then the third angle of the triangles have to be same,because the sum of angles in a triangle is $$\displaystyle{180}^{{\circ}}$$. Therefore, the triangles are similar by AA rule if two angles are same.
Calculation:
It is given that DEFG is a square.
Consider $$\displaystyle\triangle{A}{D}{G}{\quad\text{and}\quad}\triangle{G}{C}{F}$$.
$$\displaystyle\angle{A}{D}{G}=\angle{G}{C}{F}$$ both are $$\displaystyle{90}^{{\circ}}$$.
And $$\displaystyle{G}{F}{\mid}{\mid}{A}{B}$$, thus, $$\displaystyle\angle{C}{G}{F}=\angle{G}{A}{D}$$ as they are alternate angles.
Therefore, $$\displaystyle\triangle{A}{D}{G}\sim\triangle{G}{C}{F}$$ by AA rule.
Conclusion:
Hence, it is proved that $$\displaystyle\triangle{A}{D}{G}\sim\triangle{G}{C}{F}$$.

Relevant Questions

Proving Similarity in the figure DEFG is a square. Prove the following:
$$\displaystyle\triangle{A}{D}{G}\sim\triangle{F}{E}{B}$$
Given:
The given figure is,
Proving Similarity in the figure DEFG is a square. Prove the following:
$$\displaystyle{D}{E}=\sqrt{{{A}{D}\cdot{E}{B}}}$$
Given:
The given figure is,
Proving Similarity in the figure DEFG is a square. Prove the following:
Given:
The given figure is,
To prove: The similarity of $$\displaystyle\triangle{N}{W}{O}$$ with respect to $$\displaystyle\triangle{S}{W}{T}$$.
Given information: Here, we have given that NPRV is a parallelogram.
To prove : The similarity of $$\displaystyle\triangle{N}{R}{T}$$ with respect to $$\displaystyle\triangle{N}{S}{P}$$.
Given information: Here, we have given that $$\displaystyle\overline{{{S}{P}}}$$ is altitude to $$\displaystyle\overline{{{N}{R}}}\ {\quad\text{and}\quad}\ \overline{{{R}{T}}}$$ is altitude to $$\displaystyle\overline{{{N}{S}}}$$.
To prove: The similarity of $$\displaystyle\triangle{B}{C}{D}$$ with respect to $$\displaystyle\triangle{F}{E}{D}$$.
Given information: Here, we have given that $$\displaystyle\overline{{{A}{C}}}\stackrel{\sim}{=}\overline{{{A}{E}}}\ {\quad\text{and}\quad}\ \angle{C}{B}{D}\stackrel{\sim}{=}\angle{E}{F}{D}$$
Proving Similarity In the figure CDEF is a rectangle. Prove that $$\displaystyle\triangle{A}{B}{C}\sim\triangle{E}{B}{F}$$.
Given:
The given figure is,
To check: whether the additional information in the given option would be enough to prove the given similarity.
Given:
The given similarity is $$\displaystyle\triangle{A}{D}{C}\sim\triangle{B}{E}{C}$$

The given options are:
A.$$\displaystyle\angle{D}{A}{C}{\quad\text{and}\quad}\angle{E}{C}{B}$$ are congruent.
B.$$\displaystyle\overline{{{A}{C}}}{\quad\text{and}\quad}\overline{{{B}{C}}}$$ are congruent.
C.$$\displaystyle\overline{{{A}{D}}}{\quad\text{and}\quad}\overline{{{E}{B}}}$$ are parallel.
D.$$\displaystyle\angle{C}{E}{B}$$ is a right triangle.