Honors Geometry
Chapter 4: Congruent Triangles
4-1: Congruent Figures
*Definitions:
Congruent Polygons have ______________________ corresponding parts — their matching
___________ and their matching _________________. When you name congruent polygons,
you must list the corresponding vertices ________________________________.
Theorem 4-1: The Third Angles Theorem
If two angles of one triangle are equal to two angles of another triangle, then
____________________________________________________________________________.
*Note: Be careful… This does not mean that the triangles are congruent!
4-2: Triangle Congruence by SSS and SAS
*Definitions:
The term Included means ________________________________.
*This will be an important term as we move forward!!
Postulate 4-1: The Side-Side-Side (SSS) Postulate
If three sides of one triangle are equal to __________________________________________
_________________________, then the ___________________________________________.
Postulate 4-2: The Side Angle Side (SAS) Postulate
If two sides and an included angle of one triangle are equal to _________________________
___________________________, then the two triangles are __________________________.
How do we write out a proof?
Step 1. Mark your picture based upon the given information and anything else that you can
conclude from the picture (Reflexive, Vertical Angles, etc…).
Step 2. Come up with a plan, either SSS or SAS, to show why the triangles are, in fact,
congruent.
Step 3. Start your two-column proof by writing the Given followed by any steps that are a
natural consequence of the Given information. (Definition of Midpoint, Definition of
Bisect, etc…).
Step 4. Write the next given and following the same action.
Step 5. Write down any extra parts that you came up with on your own based upon the
picture and why it’s true.
Step 6. State that the triangles are congruent and why (SSS or SAS).
EX 1:
Given: AC = AD; DB = CB
C
Prove: ∆ ABC ≅ ∆ ABD
B
A
Statements
Reasons
1. AC = AD
1. __________________________
2. AB = AB
2. __________________________
3. ________________________
3. Given
4. ∆ ABC ≅ ∆ ABD
4. __________________________
D
H
EX 2: Given: ∠1 and ∠2 are vertical angles.
Prove: ∆ GHI ≅ ∆ GFE
G is the midpoint of HF
E
EG = IG
I
2
1 G
F
Statements
Reasons
1. ∠1 and ∠2 are vertical angles
1. ________________________
2. ________________________
2. Vertical Angles are Equal.
3. ________________________
3. Given
4. ________________________
4. A midpoint cuts a segment into 2 equal parts.
5. EG = IG
5. ________________________
6. ∆ GHI ≅ ∆ GFE
6. ________________________
4-3: Triangle Congruence by ASA and AAS
Postulate 4-3: The Angle-Side-Angle (ASA) Postulate
If two angles and an included side of one triangle are equal to _________________________
____________________________________________________________________________,
then the triangles are _______________________.
Theorem 4-2: The Angle-Angle-Side (AAS) Theorem
If two angles and a non-included side of one triangle are equal to ______________________
____________________________________________________________________________,
then the triangles are _______________________.
Congruence Postulates & Theorems:
1)
3)
2)
4)
*What are not possible postulates to prove triangle congruency?? __________ & __________
EX 1: Name that postulate or theorem!
C
A
(((( ≅ ((((
EX 2: Given: ∡A ≅ ∡C; BE
BD
Prove: △ ABE ≅ △ CBD
B
E
EX 3: Given: ∠ M = ∠ K, WM // RK
D
M
R
W
K
Prove: ∆ WMR ≅ ∆ RKW
4-4: Using Corresponding Parts of Congruent Triangles
*Definitions:
If two triangles are congruent, their ______________________________________________.
Otherwise known as: ______________________!
C
means: _________________________________________
P
means: _________________________________________
C
means: _________________________________________
T
means: _________________________________________
E
means: _________________________________________
*Once we show that the triangles are congruent that means ALL of their corresponding
Parts are equal.
C
EX:
,,,,,⃗ Bisects ∠CAD
Given: AC = AD; AB
Prove: ∠C = ∠D
B
A
Statements
Reasons
1. AC = AD
1. Given
2. ,,,,,⃗
AB bisects ∠CAD
2. Given
3. ________________________
3. If a ray bisects an angle, it divides into two
equal angles.
4. ________________________
4. Reflexive
5. ∆ ABC ≅ ∆ ABD
5. _______________________
6. ∠C = ∠D
6. _______________________
D
4-5: Isosceles and Equilateral Triangles
*Definitions:
Isosceles triangles are common in the real world. You can frequently see them in structures
such as bridges, and buildings, as well as in art and design.
The congruent sides of an isosceles triangle are its ___________.
The third side is the ________________.
The two congruent legs form the _____________ _________________.
The other two angles are the ____________ _______________.
Theorem 4-3: The Isosceles Triangle Theorem
If two sides of a triangle are equal, then the ________________________________________
___________________.
Theorem 4-4: The Converse of the Isosceles Triangle Theorem
If two angles of a triangle are equal, then the _______________________________________
___________________.
Theorem 4-5
If a line bisects the vertex angles of an isosceles triangle, then the line is also the
__________________________________ of the __________________.
*Definition:
A Corollary is a ___________________________________________________
________________________________________________________________.
Corollary to Theorem 4-3:
If a triangle is equilateral, then it is also ___________________________________________.
Corollary to Theorem 4-4
If a triangle is equiangular, then it is also __________________________________________.
Let’s find the measures of the missing angles in each triangle.
40°
42°
53°
75°
4-6: Congruence in Right Triangles
*Definitions:
In a right triangle, the side opposite the right angle is called the _______________________.
It is always the ____________________ side in any right triangle. The other two sides that
form the right angle of the triangle are called the __________________.
Theorem 4-6: The Hypotenuse-Leg (HL) Theorem
If the hypotenuse and a leg of one right triangle are equal to the _______________________
and _____________ of another _________________ _____________________, then
______________________________________________________.
*Note: Three conditions must be met in order use the HL Theorem!
1)
We must have two _________________ triangles.
2)
We must have two equal ______________________________.
3)
We must also have a pair of ______________ legs.
EX:
D
B
((((; AB
(((( ⊥ ((((
Given: ((((
BE bisects AD
BC; ((((
DE ⊥ (((
EC(; ((((
AB ≅ ((((
DE
Prove: △ ABC ≅ △ DEC
(7 Steps!)
A
C
E
4-7 “Congruence in Overlapping Triangles”
Sometimes we have triangles that _______________________ each other. When triangles overlap each
other they often will have a common _____________________ or a common ___________________
that will help us prove that we have congruent triangles.
*Hint: First- we should re-draw both of the triangles that overlap, then find the common angle or side.
Example: (4-steps)
Z
Y
Given: ∠ZXW ≅ ∠YWX and ∠ZWX ≅ ∠YXW
Prove: ZW ≅ YX
W
X
L
Example: (5-steps)
I
E
Given: ∠L ≅ ∠S and RE ≅ RH
Prove: LH ≅ SE
S
H
R