Download Triangle Congruency and SAS

Name ________________________
Worksheet 2.3
Triangle Congruency and SAS
1. Review of Triangle Terminology. Write a robust definition (in conditional form) of the
following and draw a labeled sketch. Recall definitions are bi-conditional.
Name
Definition
Sketch
D
Scalene Triangle
If a triangle has no congruent
sides, then it is scalene.
E
F
Isosceles Triangle
Equilateral Triangle
Acute Triangle
Obtuse Triangle
Right Triangle
2. Which of the following triangles are possible? Draw a sketch of all possible triangles.
a. An obtuse isosceles triangle
b. A right equilateral triangle
c. A scalene acute triangle
d. An isosceles right triangle
3. Congruent Triangles. Since triangles are simply enclosed figures made up of only three sides
and three angles, then the following definition should follow:
Definition: If each of three sides of a triangle is congruent to the corresponding sides of
another triangle and the corresponding angles are also congruent, then the triangles are
congruent.
Suppose we are given two triangles (such as the ones below):
W
B
C
U
V
A
Specifically, we know ABC  WVU if all corresponding angles and corresponding sides
are congruent.
To prove that all sides and angles are congruent every time we want to prove two triangles
are congruent would be needlessly time consuming. We therefore need to understand that
there are certain MINIMUM conditions which guarantee congruence. We will study these
minimum conditions in their entirety, but we will begin by looking at two of these scenarios,
each stated as a postulate:
Side-Angle-Side (SAS): -- Postulate
If two sides of one triangle are congruent to two sides of a second triangle AND the included
angles are also congruent, then the triangles are congruent. Place tick marks for this
scenario.
C
Given:
AB  DH
A   D
CA  DG
D
A
B
G
H
Therefore:

_______________


________________ by the SAS postulate.
(Be sure that corresponding vertices are correct in your statement)
4. Practice. Use the given information to complete each statement. If triangles cannot be
shown to be congruent from the information given, write “cannot be determined.”
a.
C is the midpoint of BE & AD.
ABC  
Reason (briefly explain in your own words);
b.
KI  IT ; IE bisects TEK
KIE  
Reason (briefly explain in your own words);
5. Extension from Congruent Triangles. Suppose we look at a scenario where we have been
given or have shown that AB  WV , B  V , and BC  UV . Make tick markings on the
triangles below that reflect this information.
W
B
C
U
V
A
a. What conclusion can we draw about these triangles and why?
b. Write down all the conclusions that can be drawn concerning all remaining
corresponding parts of these triangles.
c. The justification we will use for the statements in part b is the acroynm C P C T C.
What do you think CPCTC stands for?
6. Complete the proof (once you prove the first statement, simply continue your proof to reach
the second statement).
Q
Given: AQ  BC
1  2
AP  AR
Prove:
P
1) APQ  ARQ
2) AQP  AQR
1
B
Statements
R
2
A
C
Reasons
Please notice that, in order to use CPCTC in your proof, you need to have a statement
about congruent triangles listed in your proof BEFORE the CPCTC step!
7. Write a valid proof involving a pair of congruent triangles. Remember, to prove congruent
triangles you need THREE pieces of information regarding congruent parts (hence, SAS).
S
Given: SZ  TX
SY  TY
T
Y
Prove: SX  TZ
X
Statements
Z
Reasons
8. Consider a triangle whose sides measure 2t 1 , t  5 , and 3t  8 meters.
a. Determine all possible value(s) for t , making the triangle isosceles.
b. Is it possible to find a value for t which will make the triangle equilateral? If so,
determine its value.