Proving Similarity In the figure CDEF is a rectangle. Prove that

Given:

The given figure is,

Tahmid Knox
2020-11-01
Answered

Proving Similarity In the figure CDEF is a rectangle. Prove that

Given:

The given figure is,

You can still ask an expert for help

Pohanginah

Answered 2020-11-02
Author has **96** answers

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$180}^{\circ$ . Therefore, the triangles are similar by AA rule if two angles are same.

Proof:

It is given that CDEF is a rectangle.

So,$\mathrm{\angle}C=\mathrm{\angle}EFB$ .

$\mathrm{\angle}B$ is common in $\mathrm{\u25b3}ABC{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\mathrm{\u25b3}EBF$ .

Therefore, the two triangles are similar by AA rule.

Thus,$\mathrm{\u25b3}ABC\sim \mathrm{\u25b3}EBF$ .

Similarly,$\mathrm{\angle}C=\mathrm{\angle}ADE$ .

$\mathrm{\angle}A$ is common in $\mathrm{\u25b3}ABC{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\mathrm{\u25b3}AED$ .

Therefore, the two triangles are similar by AA rule.

Thus,$\mathrm{\u25b3}ABC\sim \mathrm{\u25b3}AED$ .

Since,$\mathrm{\u25b3}ABC\sim \mathrm{\u25b3}EBF{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\mathrm{\u25b3}ABC\sim \mathrm{\u25b3}AED,Thus,\mathrm{\u25b3}EBF\sim \mathrm{\u25b3}AED$ .

Therefore,$\mathrm{\u25b3}ABC\sim \mathrm{\u25b3}AED\sim \mathrm{\u25b3}EBF$ .

Conclusion:

Hence, it is proved that$\mathrm{\u25b3}ABC\sim \mathrm{\u25b3}AED\sim \mathrm{\u25b3}EBF$ .

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

Proof:

It is given that CDEF is a rectangle.

So,

Therefore, the two triangles are similar by AA rule.

Thus,

Similarly,

Therefore, the two triangles are similar by AA rule.

Thus,

Since,

Therefore,

Conclusion:

Hence, it is proved that

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