Question

Can someone explain?

Answer 1

Explanation:

Statement 2:

  Angle J is congruent to itself

Reason 2:

  Reflexive property of congruence

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Statement 3:

  ΔHIJ ~ ΔGHJ

Reason 3:

  SAS similarity theorem

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The sides given as proportional (having the same ratio) are corresponding sides in the two triangles. The first pair of sides (HJ, GJ) are named by the first and last letters of the triangle names, so correspond. The second pair of sides (IJ, HJ) are named by the last two letters of the triangle names, so correspond.

The angle between these corresponding sides is the one at the vertex whose name is the point shared by the sides. In the first triangle, the two sides of interest are HJ and IJ, which share the point at J. Thus angle J is the angle between these two sides. In the second triangle, the two sides of interest are GJ and HJ, which share the point at J. Hence angle J is the angle between these two sides, also.

So, we have corresponding sides that are proportional and the angle between them that is congruent (to itself). This allows us to invoke the SAS theorem for triangle similarity.