Download Additional Congruence and Applications

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Worksheet 2.4
Additional Congruence Conditions & Application of Congruence
1. Triangle Congruence Postulate. We have seen that the SAS postulate is helpful in proving
congruent triangles. As you can believe after a little thought, if two triangles have all three
sides congruent, then the triangles must be congruent. We will take this as the SSS (sideside-side) postulate and use it without proof, though it is often proved through other means
and thus considered a theorem.
C
Given:
AB  DH
CA  DG
CB  GH
D
A
B
G
H

 _______________
 ________________ by SSS.
Therefore:
(Be sure that corresponding vertices are correct in your statement)
2. Yet Another Congruence Postulate. Just as with SSS, as you can believe after a little thought,
if two triangles have any two sets of corresponding angles congruent and their included
corresponding sides, then the triangles must be congruent. We will take this as the ASA
(angle-side-angle) postulate and use it without proof, though it is often proved through other
means and thus considered a theorem.
Given:
AB  JM
B  M
A  J

 _______________
 ________________ by ASA.
Therefore:
(Be sure that corresponding vertices are correct in your statement)
3. Now consider an isosceles triangle where DE  EF , such as the
one provided:
E
a. Make a conjecture about other parts of this triangle:
D
F
b. Use the space below to prove your conjecture.
E
Given:
D
Prove:
F
Statements
Reasons
d. Clearly state the theorem you just proved in conditional (if/then) form:
Theorem:
4. Recall that the converse of a theorem is not guaranteed to be true.
a. State the converse of the theorem from question 3:
b. If you think your statement in part a is false, explain why. If you think it is true, outline a
proof (but you do not need to write a formal proof).
5. Suppose you are given two angles that are both supplementary and congruent. Sketch a
diagram of this scenario. Make a conjecture about these angles and then algebraically prove
your conjecture.
Now complete the following statement, which we will find to be a very useful theorem:
Theorem: If two angles are supp. and congruent, then ___________________________.
Proofs: Make a brief outline or plan before starting each proof.
6. Outline/Plan:
O
Given: OP  MR
3  4
M  R
P is the midpt. of MR
N
Prove: N  Q
4
3
M
Statements
Q
P
Reasons
R
7. Outline/Plan:
Given: 2  5
V
3  4
2
3
W
Y
Prove: WY bisects VWX
5
4
X
Statements
Reasons
8. The Perpendicular Bisector Theorem
A point is on the perpendicular bisector of a segment if and only if it is equidistant from the
the endpoints of the segment.
a. Prove this theorem “in one direction” (stated below) using triangle congruence.
If a point is on the perpendicular bisector of a segment, then it is equidistant from the
endpoints of the segment.
Given:
Prove:
Statements
Reasons
b. Write down the converse of the statement given above. We will be proving this at a later
date, once we have some additional tools at our disposal.
9.
Outline/Plan:
Given: AC  DB
ACB  DBC
Prove: AED is isosceles
Statements
Reasons