Question 548932: I'm having a little trouble with trying to figure this problem out
Given: Isos. Triangle STU
ST is congruent to TU
V is the midpoint of SU
Prove: a)triangle svt is congruent to triangle uvt
B)tv is perpendicular to su
C)tv bisects angle stu
D) angle s is congruent to angle u
All I can get is:
ST is congruent to TU:Given
V is the midpoint of SU:Given
SV is congruent to VU:Def. Of midpoint
I'm stuck on what to do next.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You said:
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ST is congruent to TU: Given <--- Good
V is the midpoint of SU: Given <--- Good
SV is congruent to VU: Def. Of midpoint <--- Good
I'm stuck on what to do next.
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If you can prove part A) triangle SVT is congruent to triangle UVT, then proving B), C), and D) can be done using the properties of congruent triangles.
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There are several ways you can prove part A). Here's one:
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You have already said ST and TU are congruent.
Furthermore, you have correctly said SV is congruent to VU by definition of midpoint.
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Now note that TV is a common side in the triangles STV and UTV. And TV is congruent to TV: Identity
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All you need to do now is say that triangle STV is congruent to triangle UTV using side-side-side congruent to side-side-side.
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Note that you can now say that D) is true, Angle S is congruent to Angle U because corresponding parts of congruent triangles are congruent.
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Or another way you could have proven A) is to also to begin with the statements you have already made:
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ST congruent to UT: Given
V is the midpoint of SU: Given
SV is congruent to VU: Def. Of midpoint
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Then add the following (presuming you already have studied these properties):
Triangle STV is isoseles: Given
You have already said ST congruent to UT: Given. So you can add that
Angle S congruent to Angle U: Angles opposite of congruent sides in isosceles triangles are congruent. [However, this is item D) you are to prove and you are just stating a reason for item D). So probably the best way to do this problem is to use the side-side-side method and then use the results of that to prove D).]
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And now you can say triangle STV is congruent to triangle UTV using side-angle- side congruent to side-angle-side using ST-Angle S-SV congruent to UT-Angle U-UV
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Now that you have demonstrated A) you can go on to B) TV is perpendicular to SU by noting the following using appropriate words that your instructor will accept:
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SU is a straight line (180 degrees)
Angle SVU congruent to Angle UVT:Corresponding parts of congruent triangles
mAngle SVU + mAngle UVT = 180 degrees
Substitute mAngle SVU for its congruent counterpart mAngle UVT to get:
mAngle SVU + mAngle SVU = 180
2 * mAngle SVU = 180
mAngle SVU = 180/2 = 90
If mAngle SVU = 90 then its congruent mAngle UVT also equals 90
Therefore TV perpendicular to SU: definition of perpendicular.
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On to C)TV bisects Angle STU.
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Angle STV is congruent to Angle UTV: Corresponding parts of congruent triangles
Therefore, TV bisects Angle STU: definition of angle bisector.
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And finally D)Angle S is congruent to Angle U.
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If you used the side-side-side congruent to side-side-side procedure for showing that triangle STV is congruent to triangle UTV then you can immediately say that Angle S is congruent to Angle U because corresponding parts (both angles and sides) of congruent triangles are congruent.
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Hope this gives you enough information so that you can wade your way through this problem. You were off to a very good start with what you had done already.
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