Name: _______________________
Date: ___________________
Period: _____
Congruent Triangles
______________ = geometric figures have exactly the same shape and size.
_____________________ = all _________________ (or matching parts) are
congruent within 2 or more polygons. These include:
-
____________________
-
____________________
∴ Two polygons are ____________ if and only if their ________________ are
________________.
Example: corresponding angles
∠A≅_____
∠B≅______
∠C≅_____
Corresponding sides
_____ ≅ HJ
______ ≅ JK
______ ≅ HK
Congruence Statement
ΔABC ≅ Δ________
ΔBAC ≅ ΔHJK
Label the corresponding parts of each pair of congruent polygons.
Remember?
Conditional statement
if ____, then ____
Converse statement
if ____, then ____
if true we say “___________”
↳
●
●
●
_____________: If 2 polygons are congruent, then their corresponding
parts are congruent
____________: If the corresponding parts are congruent, then the 2
polygons are congruent
○ ஃ _____________if 2 polygons are congruent, then their
corresponding parts are congruent
For triangles, corresponding parts of congruent triangles are congruent
(__________)
________________________
If 2 angles of 1 triangle are __________ to 2
angles of a second triangle, then the third angles of
the 2 ___________ are congruent.
Example: If ∠C ≌ ______ and ∠B ≌ _______,
then ∠A ≌ ________
Properties of Triangle Congruence
____________ Property of Triangle Congruence
△ABC ≌ △ABC
______________ Property of Triangle Congruence
If △ABC ≌ △EFG, then △EGF ≌ △ABC
_____________ Property of Triangle Congruence
If △ABC ≌ △EFG and △EGF ≌ △JKL, then △ABC ≌ △JKL
Write a two-column proof:
Given: DE ≌ GE, DF ≌ GF, ∠D ≌ ∠G,
∠DFE ≌ ∠GFE
Prove: △DEF ≌ △GEF
Statements
1.
Reasons
DE ≌ GE, DF ≌ GF
2. EF ≌ EF
3. ∠D ≌ ∠G, ∠DFE ≌ ∠GFE
4. ∠DEF ≌ ∠GEF
5. △DEF ≌ △GEF
Write a two-column proof.
Given: ∠J ≌ ∠P, JK ≌ PM,
JL ≌ PL, and L bisects KM
Prove: △JLK ≌ △PLM
Statements
1.
∠J ≌ ∠P, JK ≌ PM, JL ≌ PL, and L
bisects KM
2. JKL ≌ PLM
3. LK ≌ LM
4. ∠K ≌ ∠M
5. △JLK ≌ △PLM
Reasons
Name: _______________________
Date: ___________________ Period: _____
Congruent Triangles
Congruent = geometric figures have exactly the same shape and size.
Congruent polygons = all corresponding parts (or matching parts) are congruent
within 2 or more polygons. These include:
-
Corresponding angles
-
Corresponding sides
∴ Two polygons are congruent if and only if their corresponding parts are
congruent
Example: corresponding angles
∠A≅∠H ∠B≅∠J ∠C≅∠K
Corresponding sides
AB ≅ HJ BC ≅ JK
AC ≅ HK
Congruence Statement
ΔABC ≅ ΔHJK
✔
ΔBAC ≅ ΔHJK
Label the corresponding parts of each pair of congruent polygons.
x
Remember?
Conditional statement
if p, then q
Converse statement
if q, then p
if true we say “if and only if”
↳
●
●
●
Conditional: If 2 polygons are congruent, then their corresponding parts are
congruent
Converse: If the corresponding parts are congruent, then the 2 polygons are
congruent
○ ஃ If and only if 2 polygons are congruent, then their corresponding
parts are congruent
For triangles, corresponding parts of congruent triangles are congruent
(CPCTC)
Third Angle Theorem
If 2 angles of 1 triangle are congruent to 2 angles of
a second triangle, then the third angles of the 2
triangles are congruent.
Example: If ∠C ≌ ∠K and ∠B ≌ ∠J,
then ∠A ≌ ∠L
Properties of Triangle Congruence
Reflexive Property of Triangle Congruence
△ABC ≌ △ABC
Symmetric Property of Triangle Congruence
If △ABC ≌ △EFG, then △EGF ≌ △ABC
Transitive Property of Triangle Congruence
If △ABC ≌ △EFG and △EGF ≌ △JKL, then △ABC ≌ △JKL
Write a two-column proof:
Given: DE ≌ GE, DF ≌ GF, ∠D ≌ ∠G,
∠DFE ≌ ∠GFE
Prove: △DEF ≌ △GEF
Statements
1.
DE ≌ GE, DF ≌ GF
Reasons
Given
2. EF ≌ EF
Reflexive Property of Congruence
3. ∠D ≌ ∠G, ∠DFE ≌ ∠GFE
Given
4. ∠DEF ≌ ∠GEF
Third angles theorem
5. △DEF ≌ △GEF
Definition of congruent polygons
Write a two-column proof.
Given: ∠J ≌ ∠P, JK ≌ PM,
JL ≌ PL, and L bisects KM
Prove: △JLK ≌ △PLM
Statements
1.
∠J ≌ ∠P, JK ≌ PM, JL ≌ PL, and L
bisects KM
Reasons
Given
2. JKL ≌ PLM
Vertical Angles (are congruent)
3. LK ≌ LM
Definition of segment bisector
4. ∠K ≌ ∠M
Third Angles Theorem
5. △JLK ≌ △PLM
CPCTC