Download Geometry Notes TC – 1: Side - Angle

Geometry Notes TC – 1: Side - Angle - Side
Congruent Polygons
Review: Two polygons are congruent if
Also, two polygons are congruent if (and only if)
1.
2.
Ex: If ABC  PQR, then
a. All pairs or corresponding parts are congruent
b. There is a rigid motion for which the
image of ABC is PQR.
P
C
R
B
A
Q
Problem: Saying two figures are congruent if one is the image of the other under a rigid motion is a good
definition of congruence. But it is not always a convenient method to prove two figures are congruent.
Ex: Are the triangles below congruent?
If they are, then the transformation
B
C
will map ABC onto DEF.
F
D
A
E
C F
But how can we be sure that the triangles actually
map perfectly one onto the other?
A
D
E
B
Proving Two Triangles Congruent
If all three pairs of corresponding sides are congruent and
all three pairs of corresponding angles are congruent, then
two triangles must be congruent.
Is it possible to prove two triangles congruent without proving all six pairs of corresponding parts congruent? If
so, what is the least number of congruent pairs of corresponding parts we need?
One pair of sides?
One pair of angles?
Two pairs of sides?
Two pairs of angles?
One pair of each sides, angles?
Side-Angle-Side
If two sides and the included angle of one triangle are all congruent to the corresponding sides and angle of a
second triangle, then the two triangles are congruent.
Given: ABC and A'B'C'
AB  A ' B ' , AC  A ' C ' , and A  A'
B'
B
Note: A is called the included angle for sides
AB and AC because it is the angle formed by
those two sides (where those two sides meet).
C'
A'
C
Show via rigid motions that A'B'C'  ABC.
A
B'
C'''
B
C'
B"
A'
C"
C
A
Ex: Given: AB  CD , AB || CD
B
A
Prove: ABC  CDA
D
C
Geometry HW: Triangle Congruence – 1 Side Angle Side
For the following four problems, determine if the information given in the diagram is sufficient to prove the
triangles congruent and give a reason for your answer.
1. F
B
2.
D
D
C
3. B
A
A
E
A
4.
C
D
C
E
D
B
A
B
C
In the next three problems, name the pair of corresponding sides or angles that would need to be proved
congruent, in addition to the ones already shown, in order to prove the triangles are congruent by SAS.
C
A
D
A
5.
6.
7.
C
C
B
E
B
A
D
B
D
Write complete geometry proofs for the following (diagrams below):
B
8. Given: AB  AD , AC bisects BAD
Prove: ABC  ADC
C
A
D
R
9. Given: AS  RT , A is the midpoint of RT
Prove: RAS  TAS
E
F
A
T
S
P
10. Given: PQ  RS , PQ || RS , QUTS , QU  ST
Prove: PQT  RSU
T
S
Q
U
R
C
11. a. In a triangle, what is
A
B
1) an altitude?
2) a median?
3) an angle bisector?
b. Copy the diagram at right onto your own paper and on it draw and label altitude CP , median CM and
angle bisector CX .