Download Congruence Same size AND same shape. Congruent figures can be

Congruence
Same size AND same shape.
Congruent figures can be mapped onto each other such that
there intersection is the same
A as the figures themselves.
A
Corresponding Parts
are match up in a one-to-oneA correspondence.
A!D
A
AB ↔ DE
C
C ↔ EF
BC
C!F
AC ↔ DF
C
B up.
B
The vertices andBsides must
match
B
B
C
A
B!E
B
We can therefore write
C
ABC!DEF
A
E
C
point A matches up to point D
point B matches up to point E
point C matches up to point F
The order of the letters is important
D
F
Angles are congruent if they have the same measure.
Sides are congruent if they have the same length.
Congruence is therefore an equation between geometric figures, while
equality is an equation between numbers.
CONGRUENCE AND EQUALITY ARE NOT INTERCHANGEABLE!!!!!!!
!ABC ! !DBF means the angles have the same measure.
!ABC= !DBF means the angles are the same angle.
This is true for triangles as well. You can only say two triangles are
equal if they are the same triangle.
Two triangles are congruent if there is a correspondence between all
pairs of sides and all pairs of angles.
∆ABC! ∆DEF tells us 12 things at once.
AB ≅ DE, AB = DE, AC ≅ DF, AC = DF, BC ≅ EF, BC = EF
∠A ≅ ∠D, m∠A = m∠D, ∠B ≅ ∠E, m∠B = m∠E, ∠C ≅ ∠F, m∠C = m∠F
B
A
C
A side of the triangle is said to be included by the angles whose
vertices are the endpoints of the segment.
Side AC is included by !A and !C.
An angle of the triangle is said to be included by the sides of the
triangle which lie on the sides of the angle.
!C is included by sides AC and BC.
Congruence for segments is an equivalence relation.
Congruence for triangles is an equivalence relation.
A
B
C
Create ∆ABC using the following specifications.
1.  AB = 3 inches, BC = 4 inches, m!B = 45!SAS
2.  AB = 3 inches, AC = 4 inches, m!B = 45!SSA
3.  m!A = 75!, m!B = 45!, AB = 3 inches
ASA
4.  m!B = 45!, m!C = 60!, AB = 3 inches
AAS
5.  AB = 3, BC = 4, AC = 6
SSS
6.  m!A = 75!, m!B = 45!, m !C = 60!
AAA
SSS: If there exists a correspondence between
the vertices of two triangles such that three
sides of one triangle are congruent to the
corresponding sides of the other triangle,
then the two triangles are congruent.
B
SAS: If there exists a correspondence between
the vertices of two triangles such that two
sides and the included angle of one triangle
are congruent to the corresponding parts
B
of the other triangle, then the two triangles
are congruent.
ASA: If there exists a correspondence between
the vertices of two triangles such that two
angles and the included side of one triangle
are congruent to the corresponding parts
of the other triangle, then the two triangles
are congruent.
F
A
C
F
A
C
D
E
F
A
B
D
E
C
E
D
The order of the letters is important.
SAS: Every SAS correspondence is a congruence.
F
A
∆ABC ! ∆FED
B
Corresponding vertices
C
E
D
E
B
Given: ∠A ≅ ∠F, AD ≅ CF, ∠EDA ≅ ∠BCF
Pr ove: ΔABC ≅ ΔFED
A
D
C
F
What congruence should be used?
Given: A pair of congruent angles. (!A ! !F)
Therefore ASA and SAS are the only possibilities.
(
Given: A piece of a side, can you get the entire side? AD ≅ CF
Given: Another Angle. (!EDA ! !BCF)
Therefore ASA and is the only possibility.
Find the third angle of the triangle.
Brainstorm your statements.
Write the proof.
)
Side
Angle
E
B
Given: ∠A ≅ ∠F, AD ≅ CF, ∠EDA ≅ ∠BCF
Pr ove: ΔABC ≅ ΔFED
Angle
Statements
1. ∠A ≅ ∠F, AD ≅ CF, ∠EDA ≅ ∠BCF
2. AC ≅ DF
3. ∠ADC and ∠DCF are straight ∠s
4. ∠EDA and ∠EDF are sup plementary
∠BCF and ∠BCA are sup plementary
5. ∠BCA ≅ ∠EDF
6. ΔABC ≅ ΔFED
A
D
C
F
Reasons
1. Given
2. segment addition property
3. assumed from diagram
4. If the sum of 2 ∠s is a straight ∠,
then they are sup plementary
5. Supp. of ≅ ∠s are ≅
6. ASA (1,2,5)
Order is important!!