Triangles congruence pls help.

Question

Answer 1

Hi Alia M.,

So you are given a particular rule (for example, ASA; A=angle and S=side) and you need to apply = find that arrangement on each of your two triangles in your figure. So for example, with ASA, in the third figure of the problem, you see sides XW and XY are already indicated as conguent. So you need to find two other sets of angles, in order to apply the rule. But, no other sets of angles are marked as congruent. Do you throw up your hands? Not at all! (That is, you shouldn't.) There is always some other place where you could find congruent angles, typically where 2 lines cross, here at point X. Opposite angles at a line crossing are always congruent. So that gives you VXW and ZXY as congruent angles. Note that I have named these angles so as to "align" them into the conguent arrangement.

To complete your answer here, you just need to follow the "ASA" order around each of the triangles -- clockwise for triangle XWV, and counterclockwise for triangle XYZ -- both starting from point X. The next angles you get to (beyond side XW and side XY respectively, in the two triangles), need to be made congruent "the third congruence" in order to "prove the congruence".

Note that until you state that third congruence, all you "have" in the two triangles is "AS" which does NOT make (prove) them congruent. And in geometry, proof is everything -- just saying that two things might be congruent (geometry's equvalent term to "equal" elsewhere in math) isn't of much use. So if something might be congruent, or might not, you have to say "not congruent" or "not determinable" (if your teacher uses that term), in the same way as if something might be either TRUE or FALSE, you have to say "FALSE".

And in somewhat the same way, if someone might be guilty of a crime, or might not, with equal probability, in American criminal law you must deem them "Not Guilty". Only in Scottish law is there an intermediate finding, "Not Proven", which designates "likely guilty, but we can't be sure". Practical people, the Scots!

-- Cheers, - Mr. d.

Answer 2

For the first.

XY is a common side, so we know WY is congruent to XZ and XY is congruent to XY.

So for SSS we also need WX congruent to YZ.

For the second we have two pairs of congruent sides.

The included angles also need to be congruent, so we need angle A and J congruent.

For the third.

We have a pair of congruent sides.

Angles VXW and YXZ are congruent (vertical angles)

So we need angles W and Y to be congruent to apply ASA