CK-12 Geometry-Concepts

http://www.ck12.org Chapter 4. Triangles and Congruence

TABLE4.15:

Statement Reason

1. 
   
   
   
   1. 2.^6 ST Wand^6 U T Vare right angles 2.
   
   
   
   


2. 
   
   
   
   3. 4.ST∼=T V,W T∼=T U 4.
   
   
   
   


3. 4 ST W∼= 4 U T V 5.
   6.W S∼=UV 6.


4. If two right triangles have congruent hypotenuses and one pair of non-right angles that are congruent, are the two
   right triangles definitively congruent?

Answers:

1. 
   

   All we know is that two pairs of sides are congruent. Since we do not know if these are right triangles, we cannot
   use HL. We do not know if these triangles are congruent.

   
   


2. 
   

TABLE4.16:

Statement Reason

1.SV⊥WU 1. Given

2.^6 ST Wand^6 U T Vare right angles 2. Definition of perpendicular lines.
3.Tis the midpoint ofSVandWU 3. Given
4.ST∼=T V,W T∼=T U 4. Definition of midpoint
5. 4 ST W∼= 4 U T V 5. SAS
6.W S∼=UV 6. CPCTC

Note that even though these were right triangles, we did not use the HL congruence shortcut because we were not
originally given that the two hypotenuses were congruent. The SAS congruence shortcut was quicker in this case.

3. Yes, by the AAS Congruence shortcut. One pair of congruent angles is the right angles, and another pair is
   given. The congruent pair of sides are the hypotenuses that are congruent. Note that just like in #2, even though
   the triangles are right triangles, it is possible to use a congruence shortcut other than HL to prove the triangles are
   congruent.

Explore More

Using the HL Theorem, what information do you need to prove the two triangles are congruent?