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Using the law of reflection and Geometry, prove that image distance equals object...

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Using the law of reflection and Geometry, prove that image distance equals object distance for a plane mirror.draw horizontal rays from the flame&base

Of a candle. Draw a ray from the flame that meets the mirror at the axis.

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You need to sketch the mirror as a vertical line and you need to place an object in front of this vertical line (to the left side).

You need to remember that the angle between the normal to the mirror at reflection point and the incident light is equal to the angle between the normal line and the reflected light.

The reflected light is the light an observer could see. 

You need to consider two incident lights that go from object and hit the mirror at reflection points.

These two incident lights will give back two reflected ligths. You need to extend these reflections back behind the mirror. These extensions will meet at a point that represents the location of the image.

The angle made by reflected light to normal is equal to the angle made by extension of reflected light to normal.

Since this angle is congruent to the angle between normal and incident light, then the angle between the incident light and mirror and the angle between the extension and mirror are equal.

The incident lights and the mirror form a triangle that is congruent to triangle that is formed by extensions of reflected lights and mirror having two congruent angles and one common side (mirror).

The distance from object in front of mirror to mirror is the height of triangle and it is congruent to the distance from image behind mirror to mirror, by proven congruence of triangles.

Hence, using congruent triangles yields that object distance is equal to image distance.

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