Download 4.1: Congruent Figures Congruent Polygons: Corresponding Angles

4.1: Congruent Figures
Congruent Polygons:
Corresponding Angles:
Corresponding Sides:
Congruence Statement:
Naming Polygons:
1) List the vertices by starting at one and working around (the corners of the figure)
2) If it’s a triangle use a small triangle in front of the vertices, for anything else,
just use the word polygon
Example: CAT  JSD. List each of the following.
1. three pairs of congruent sides
2. three pairs of congruent angles
Examples: Each pair of polygons is congruent. Find the measures of the numbered angles.
3.
4.
How can you show that two figures are congruent?
Example: Can you conclude that the triangles are congruent. If so, write a congruence
statement.
5. FGH and JKH
Third Angle Theorem:
Proof of the Third Angles Thm:
Given: A  D , B  E
Prove: C  F
A
B
D
C
E
Examples: Find the values of the variables.
6.
7.
Examples: ABCD  FGHJ. Find the measures of the given angles or lengths of the given sides.
8. mB = 3y, mG = y + 50
Example 9:
Given: BD is the angle bisector of ABC.
BD is the perpendicular bisector of AC .
Prove: ADB  CDB
F
4.2: Triangle Congruence by SSS and SAS
Parts of a Triangle
1) Vertex:
2) Included Side:
3) Included Angles:
Proving Triangles Congruent:
If only a certain amount of information is given:
1) Apply the Side-Side-Side Postulate
2) Apply the Side-Angle-Side Postulate
Side-Side-Side Postulate
B
A
K
C J
L
Side-Angle-Side Postulate
B
A
K
C J
L
Draw MGT. Use the triangle to answer the questions below.
1. What angle is included between GM and MT ?
2. Which sides include T ?
3. What angle is included between GT and MG ?
Would you use SSS or SAS to prove the triangles congruent? If there is not enough
information to prove the triangles congruent by SSS or SAS, write not enough information.
Explain your answer.
4.
5.
7.
8.
9.
10. Given: BC  DC, AC  EC
Prove: ABC  EDC
Statements
11. Given: WX YZ ,WX  YZ
Prove: WXZ  YZX
6.
Reasons
4.3: Triangle Congruence by ASA and AAS
Recall:
Included Angle: Angle formed by a pair of sides
Included Side: Side connecting two angles
Angle-Side-Angle Postulate
A
X
What else would you need to know to prove the
triangles congruent by ASA?
Given:
A  X
B
C
Y
Z
Angle-Angle-Side Theorem
A
B
X
C
Y
Show on the pair of triangles one possibility of two
angles that could be congruent and the included side
that would need to be congruent.
Z
Example: Are the two triangles congruent? If so, make a congruence statement.
A
B
A
AC  BD
CAbi sec tsBAD
C
D
B
C
D
Example: Given: A  D Prove:
C   F
ABC  DEF
A
BC  EF
B
C
D
E
Example: Given: BD  BC
F
A
C
AD EC
Prove:
ABD  EBC
B
D
E
4.4: Using Corresponding Parts of Congruent Triangles
CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
If you can prove that two triangles are congruent, then you can prove any pair of corresponding parts is
congruent.
Your goal is to prove that two triangles are congruent!
What have we used to prove that triangles are congruent?
Example: State why the two triangles are congruent. Then list all other pairs of corresponding parts.
Example:
Given:
Prove: LHJ  JKL
H
J
L
Example:
Given: A is the midpoint of MT & SR
Prove: M  T
K
M
R
A
S
T
D
Example:
Given: 1  2; 3  4
Prove:
BCE  DCE
C
2
1
4
3
E
B
A
4.5: Isosceles, Equilateral, and Right Triangles
Isosceles:
Legs:
Base:
Base Angles:
Vertex Angle:
Example: Label the legs, the base angles and the vertex in the triangle. Use symbols to show which parts
are congruent.
Isosceles Triangle Theorem: If two sides of a triangles are congruent, then
Converse of the Isosceles Theorem: If two angles of a triangle are congruent, then
X
Given:
XY  XZ
XB
1 2
bi sec ts YXZ
Prove: Y  Z
Y
B
Z
Theorem 4-5: If a line bisects the vertex angle of an isosceles triangle, then the line is also perpendicular
bisector of the base.
Diagram:
Corollary: If a triangle is equilateral, then it’s equiangular.
Corollary: If a triangle is equiangular, then it’s equilateral.
Examples: Complete each statement. Explain why it is true.
1. AB 
2. BDE 
3. CBE 
 BCE
Example: Solve for x and y.
1)
y
12
2)
3)
50 50
y x
2x + 8
50
y
x
40
4)
5)
4.6: Congruence in Right Triangles
Right Triangle:
Hypotenuse:
Leg:
Diagram: Label the Hypotenuse and Leg in the Right Triangle Below
Hypotenuse – Leg Thm. (HL): If the hypotenuse and leg of one right triangle are congruent to the
hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.
Conditions For Using HL:
1) The two triangles must be
.
2) The hypotenuses must be
.
3) There is one pair of congruent
.
Example: Complete the proof.
Given: V and W are right angles ; WZ  VX
Prove: WVZ  VWX
Statements
Reasons
1)
1) Given
2)
2) Given
3)
3) Reflexive Property of Congruence
4)
4) HL Theorem
Examples: For what values of x and y are the triangles congruent by HL?
1)
2)
Examples: What additional information would prove each pair of triangles congruent by the
Hypotenuse-Leg Theorem?
3)
4)
Practice with Proofs
A
Given: AB  DC , AB  CD
Prove: ABC  CDA
B
D
Statements
C
Reasons
Q
K
A
Given: BK  BA, QB bisects KBA
Prove: KQ  AQ
Statements
B
Reasons
Given: P  T , R is the midpoint of PT
Prove: PQR  TSR
P
S
R
Q
T
Statements
Reasons
Q
K
Given : QK  QA, QB bisectsKQA
A
B
Prove : KB  AB
Statements
Reasons