Download Class Notes Triangle Congruence

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 Notes TC – 2: Angle - Side - Angle
Proving Two Triangles Congruent, Continued
Angle-Side-Angle
Theorem: If two angles and the included side of one triangle are all congruent to the corresponding angles and
side of a second triangle, then the two triangles are congruent.
Given: ABC and A'B'C'
A  A', A  A'¸ and AB  A ' B '
Note: AB is called the included side for A
and B because it is the side joining the
vertices of those two angles.
B'
B
C'
A'
C
Show via rigid motions that A'B'C'  ABC.
A
B'
C'''
B
C'
B"
A'
C"
C
A
Ex: Given: AC bisects BD at E, AB  BD , CD  BD
Prove:
C
a. ABE  CDE
b. A  C
B
A
E
D
Ex: Given: CD bisects ACB, CD is an altitude of ABC,
C
Prove: AC  BC
A
D
B
Geometry Notes TC – 3: Side - Side - Side
Proving Two Triangles Congruent, Continued
Side-Side-Side
Theorem: If all three sides of one triangle are congruent to the corresponding sides of a second triangle, then
the triangles are congruent.
B'
Given: ABC and A'B'C'
AB  A ' B ' , BC  B ' C ' ¸ and CA  C ' A '
Show via rigid motions that A'B'C'  ABC.
A'
C'
C'
B
C
A
B'
A'
C'
C'
B
C'''
C
B"
C"
C'
A
Ex: Given: Isosceles triangle ABC with vertex angle C,
CD is a median of the triangle.
Prove: A  B
Note: Most proofs come with a diagram. If one doesn’t,
try to draw a diagram that fits the givens.
Geometry Notes TC – 4: Practice
A
Given: AB || CD , DPQB , BQ  DP ,
B
AP  DB , CQ  BD
Q
Prove: AP  CQ
P
a. Tell what’s wrong with the following proof.
Statement
1. AB || CD (S)
Reason
1. Given
2. AP  DB , CQ  BD (A)
2. Given
3. BQ  DP (S)
3. Given
4. AP  CQ
4. SAS (1, 2, 3)
b. Write a correct proof.
Statement
Reason
D
C
Geometry Notes TC – 5: Isosceles Triangle Theorem
Angle Bisectors (review)
Definition: An angle bisector
Postulate: Every angle has
Isosceles Triangle Theorem
We wish to prove: If two sides of a triangle are congruent, the angles opposite those sides are also congruent.
Complete the Given and Prove below and draw a suitable diagram.
Given:
Prove:
The plan is to bisect the vertex angle of the given triangle
and then prove the two new triangles are congruent. Fill in
missing statements and reasons (including the blanks in
statement 2) to complete the proof below.
Statement
Reason
1. Given
1.
2. Let point
be on base
such that
2.
is the angle bisector of vertex angle
3.
3.
4.
4.
5.
5.
6.
6.
Ex: Draw a suitable diagram, then give an appropriate conclusion and reason:
Given: In RAT, RA  AT
Conclusion:
Reason:
Ex: CAT, shown at right, has perimeter 52.
A
a. Find the value of x.
2y + 3z
2
x
5x
z
5y + 30
C
T
2x + 8
b. Find the values of y and z.
The converse of the Isosceles Triangle Theorem is also true. (Proof is on a later HW).
Write the converse:
Ex: Determine which two sides of HUG are congruent.
U
2x + 42
H
3x + 12
5x + 6
G
Geometry Notes TC – 6: Review/Practice
To prove two triangles congruent:
To justify corresponding parts congruent:
1.
2.
3.
4.
Coming soon to a theater near you. Don’t miss them!
D
5.
Ex: Given: DIN with DI  DN , DI  IM , DN  NM , IAON ,
IO  AN
Prove: AM  OM
I
A
O
M
N
Geometry Notes TC – 7: Angle - Angle - Side
Proving Two Triangles Congruent, Continued
Angle-Angle-Side
Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding angles and
side of a second triangle, then the triangles are congruent.
Given: ABC and DEF
A  D, C  F, and AB  DE
Prove: ABC  DEF.
A
B
Ex: Give a reason why each pair of triangles is congruent.
F
C
a.
A
B
D
E
F
C
b.
A
F
C
B
D
E
D
E
Ex: Given: Pentagon TOPAZ with diagonals TP and ZP ,
TZP is isosceles with vertex P, OTZ  TZA,
O  OPA, A  OPA
T
B
Z
A
O
Prove: OP  AP
P
Geometry Notes TC – 8: Hypotenuse - Leg
Proving Two Triangles Congruent, Continued
Hypotenuse-Leg
If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and corresponding leg of
another right triangle, then the two triangles are congruent.
Given: ABC with right angle C, DEF with right angle F,
AB  DE and BC  EF
E
B
Prove: ABC  DEF
C
F
E
D
Prove: AE || CF
B
A
Ex: Given: AB  DC , BFED ,
AB  AE , CD  CF ,
BF  DE
A
C
F
D
Geometry Notes TC – 10: Indirect Proofs
C
Ex: Theorem: If a triangle is isosceles, the altitude from the vertex angle bisects
the base (i.e., the altitude from the vertex is also a median).
Given: Isosceles triangle ABC with vertex C; CD is an altitude.
Prove: CD bisects AB
A
D
B
This is called a direct proof. It starts with the givens and then uses deductive reasoning to lead to what is to be
proved.
Prove the inverse of the theorem above:
C
Given: Scalene ABC; CD is an altitude.
Prove: CD does not bisect AB
A
D
B
Indirect proof: Start by assuming the negation (opposite) of what you want to prove.
C
Given: Triangle ABC is not isosceles; CD is an altitude.
Prove: CD does not bisect AB
Assume:
A
Ex: Given: Line AB and point P not on AB .
D
B
.P
Prove: There is only one line through P perpendicular to AB .
l