SOLUTION: Hello. We are learning Inequalities for Sides and Angles of a Triangle in class. A question asks: Suppose angle WXY = angle XYZ. Which angle or of XYZ or WXY has the greatest

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Question 389517: Hello.
We are learning Inequalities for Sides and Angles of a Triangle in class.
A question asks:
Suppose angle WXY = angle XYZ. Which angle or of XYZ or WXY has the greatest measure?
The answer the book gives is angle W. I am not sure why. Can someone please explain?
Thanks.
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Found 3 solutions by Alan3354, Edwin McCravy, MathTherapy:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose angle WXY = angle XYZ. Which angle or of XYZ or WXY has the greatest measure?
The answer the book gives is angle W. I am not sure why. Can someone please explain?
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The sketch is not to scale. Since the interior angles are equal, WX is parallel to YZ.
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You can solve for all the angles, using first the Cosine Law.
15%5E2+=+12%5E2+%2B+14%5E2+-+2%2A12%2A14%2Acos%28W%29
cos(W) = 115/336 =~ 0.34226
Do the others the same way, and the smallest cosine is the largest angle.

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!


The other tutor used trigonometry, but that is not what you
are studying here.  This is geometry, not trigonometry.


The figure is not drawn to scale at all.  We must redraw it more
nearly to scale because it is necssary that YZ which has measure 17 
to be drawn to look longer than WX, which has measure 12.  

Here it is approximately to scale:



Angle W is the largest angle in triangle WXY because it's opposite the
triangle's longest side XY.

Angle XYZ is the largest angle in triangle XYZ because it is opposite the
triangle's longest side.

Therefore the largest of the six angles in both triangles is either
angle W Or angle XYZ.

Since angle WXY = angle XYZ, we can construct a triangle
that's congruent to triangle WXY.

We can locate point P on YZ, such that YP has measure 12, same as WX.
And then draw XP. [This is why we had to redraw the figure because if
you used the drawing at the top you would have to extend YZ.]



Now triangle PYX is congruent to triangle WXY by Side-angle-side.

Therefore the green line XP has the same measure as WY, or 14.



Now we see that Angle XPY is greater than angle XYP because
in triangle XPY, Angle XPY is opposite a side of measure 15
and XYP is opposite a side of measure only 14.

Since Angle XPY has the same measure as angle W, angle W is the
greatest interior angle in either triangle WXY or triangle XYZ.

Edwin

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!
In triangle XYZ, the largest angle is angle XYZ as this angle is opposite the longest side: side XZ which has a length of 18 units.

Angle XYZ in triangle XYZ is congruent to angle WXY in triangle WXY, but while angle XYZ is the largest angle in triangle XYZ, it is not the largest angle in triangle WXY. This is because the longest side in triangle WXY is side XY (15 units), therefore making the largest angle in that triangle, angle XWY, or angle W.