Question 923700: How do I prove congruence of these 2 triangles in a 2 column proof. I am given a picture of congruent triangles IHG and IHJ. The triangles share a side. Point/angle G is on the top left and it connects along the top with a horizontal line to point H which connects again horizontally along the top to point/angle J at the top right (point H seems to be the midpoint of this line/side). At point/angle H a seemingly perpendicular vertical line extends downward and connects to point I. It basically looks like an upside down (by upside down I mean the flat part is towards the top of the paper and the point is facing down.) acute isosceles triangle with a vertical bisecter half way between point/angle G and point/angle J going down to the point/angle I, that is what the picture looks like as a whole. Individually triangle IHG is drawn with G at the top left then a horizontal side going to H then a vertical line extending down to I (it appears to be perpendicular to the horizontal line coming from G)and lastly from I a diagonal line connecting back to G. Triangle IHJ looks almost identical to IHG, only mirrored over the side HI. (I hope I described the picture well enough for you!). The givens are; line GI is congruent to JI and, angle GHI is congruent to JHI but it is not given that they are right angles. The measure is unknown, the only thing known is that they are congruent. I am asked to prove triangle IHG and IHJ congruent. I have stated the givens, I have also stated that the shared side HI congruent to itself by using the reflexive property. That is where I am stuck, I cannot prove it congruent with the hypotenuse leg theorem because it is not a right triangle. The side-side-side postulate does not work either because I would need side GH of triangle IHG to be congruent to side JH of triangle IHJ. The side-angle-side postulate would not work because only one side of the 2 pairs of congruent sides touch the angle. The angle-side-angle postulate also does not work because I do not know 2 pairs of congruent angles. And the angle-angle-side theorem also fails to work also because of the lack of 2 pairs congruent angles. I thought I had it solved because I thought H was the midpoint of GJ (because it looks like it is) but it is not given. I cannot think of any other way to progress of this proof, hope fully you can. Thank you in advance!
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! We have two right triangles, so we can use the following:
If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. (Either leg of the right triangle may be used as long as the corresponding legs are used.)
we are given that hypotenuse GI is congruent to hypotenuse JI and leg HI is congruent to leg HI
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