1
Date
12/1
Lesson
1
Topic
Congruence and Triangles
12/2
2
Congruent ∆’s by SSS
12/3
3
Congruent ∆’s by SAS and ASA
12/4
4
Congruent ∆’s by AAS
12/5
5
Congruent ∆’s by HL
12/8
6
12/9
7
12/10
8
CPCTC
Exterior Angle of Triangle
Quiz
Isosceles ∆ Proofs
12/11
9
Overlapping ∆ Proofs
12/12
10
Using Sets of Triangles
12/15
11
Review Sheet
12/16
12
Unit 6 Test
2
Lesson 1: Congruence
Warm up: State a sequence of transformations that
will map one triangle onto the other.
Are the triangles congruent? Explain why or why not.
Geometric figures are congruent if they are the same size and shape. More specifically:
Two plane figures are congruent if and only if one can be obtained from the other by some
sequence of rigid motions.
Determining if two figures are congruent:
Look at each of the examples below. Determine if there is some sequence of rigid motions
(transformations) that will allow either figure to be an image of the other. If there is such a sequence
– write it on the lines below, otherwise, state why the figures are not congruent.
Example 1:
Example 2:
_____________________________________
_____________________________________
_____________________________________
_____________________________________
Example 3:
Example 4:
_____________________________________
_____________________________________
_____________________________________
_____________________________________
3
Corresponding Parts of Congruent Triangles
Two triangles are congruent if and only if all of the angles and sides of one triangle are
congruent to the corresponding sides and angles in a second triangle

The order in which the congruence statement is written determines which sides and which
angles are congruent.
Example 1:
∆ABC and ∆DEF (right) are congruent. This would be
written as ∆ABC  ∆DEF.
In the space below, write the pairs of congruent angles and
the pairs of congruent sides.
R
X
Example 2: If ∆RST  ∆XYZ, complete the statements
below.
T
S
1. R  ______
5. T  ______
2. RS  ______
6. RT  ______
3. YZ  ______
7. ∆STR  _________
4. Y  ______
8. ∆ZYX  _________
Example 3: ∆ADC  ∆CBA, list the pairs of congruent
corresponding parts and mark them in the figure.
Example 4: ∆FUN  ∆ZAP, list the pairs of congruent corresponding parts.
Y
Z
4
Example 5: In the diagram shown, ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹
Determine and state DE=___________
Determine and state 𝑚∡𝐵 = ___________
Lesson 2: Proving Triangle Congruency Using SSS
Warm up: In the diagram shown, ∆𝐴𝑆𝑀 ≅ ∆𝐸𝑅𝑁. Determine and state the
following measures:
𝑚∡𝐸 = ______
𝑚∡𝑁 = ______
EN = _______
x = _______
y = _______
Is it necessary to show that all of the corresponding sides and all of the
Corresponding angles are congruent to show that the triangles are congruent?
SSS - If three sides of one triangle are congruent to three sides of another triangle, then the
triangles are congruent.
5
Ex 1: Are the triangles congruent by SSS?
1.
2.
3.
Ex 2: What additional congruent sides do you need in order to say the triangles are
congruent by SSS?
1.
2.
Ex 3: Given: AB  AD
CB  CD
3.
6
Prove: ABC  ADC
Ex 4: Given: T is the midpoint of PQ
PQ bisects RS at T
RQ  SP
Prove: ∆RTQ  ∆STP
P
S
T
R
Q
Ex 5: Given: WX  ZY
WY  ZX
Prove: WXY  ZYX
Ex 6: Given: AC  BC
7
CD  CE
AE  BD
Prove: ADC  BEC
8
Lesson 3: Proving Triangle Congruency Using SAS and ASA
Warm up: Determine if SSS can be used to prove the following triangles congruent.
If not, state the additional information needed to use SSS.
Let’s talk about a few more “shortcuts” to proving triangles congruent:

SAS - If two sides and the included angle of one triangle are congruent to two sides
and the included angle of another triangle, then the triangles are congruent.
Ex 1: Are the triangles congruent by SAS?
1.
2.
3.
Ex 2: What additional pairs of sides or angles do you need to have congruent in order to say
the triangles are congruent by SAS?
1.
2.
3.
9
Ex 3: Given: DE  AB
EF  BC
<E and < B are right angles
Prove: DEF  ABC
D
A
E
F
B

C
ASA - If two angles and the included side of one triangle are congruent to two angles
and the included angle of another triangle, then the triangles are congruent
Ex 1: Are the triangles congruent by ASA?
3.
Ex 2: What additional pairs of sides or angles do you need to have congruent in order to say
the triangles are congruent by ASA?
1.
2.
3.
10
Ex 3: Given: E  C
D is the midpoint of EC
ADE  BDC
Prove: ∆AED  ∆BCD
A
B
E
C
D
Mixed Practice
1. Given:
RS  ST
LT  ST
M is the midpoint of ST
Prove: RSM  LTM
2.
Given: AB  AD
AC bisects BAD
Prove: ∆ABC  ∆ADC
B
C
A
D
11
3.
Given: AB bisects CD at P
AC || DB
Prove: ACP  BDP
Lesson 4: Proving Triangle Congruency Using AAS
Warm up: Which method can be used to prove each pair of triangles congruent?
a)
b)
c)
_________________
________________
________________
AAS - If two angles and the non-included side of one triangle are
congruent to two angles and the non-included side of a second triangle,
then the triangles are congruent.
ABC  A'B'C' by AAS
Ex 1: Are the triangles congruent by AAS?
1.
2.
3.
12
Ex 2: What additional pairs of sides or angles do you need to have congruent in order to say
the triangles are congruent by AAS?
A
1.
2.
3.
B
Ex 3: Given: 5  6
X  Y
Prove: XWZ  YWZ
Ex 4: Given: P  N
M is the midpoint of ̅̅̅̅
𝐾𝑄
Prove: PMK  NMQ
C
D
13
Ex 5: Given: RA  AB
PB  AB
AB bisects RP at M
Prove: MAR  MBP
Ex 6: Given: E  B
EF || BC
DC  AF
Prove: EDF  BAC
14
Lesson 5: Proving Triangle Congruency Using HL
Warm up: Which method can be used to prove each pair of triangles congruent?
a)
b)
c)
_________________
________________
________________
Hypotenuse – Leg (HL) - If the hypotenuse and one leg of a right triangle are congruent to
the hypotenuse and leg of a second right triangle, then the two right triangles are congruent.
Note: You must show that
you have
__________________________
to use this method!
ABD  CBD by HL
Ex 1: Are the triangles congruent by HL?
1.
2.
3.
Ex 2: What additional information do you need in order to say the triangles are congruent by
HL?
1.
2.
3.
15
Ex 3: Given: ABD is a right angle
CDB is a right angle
AD  CB
Prove: ABD  CDB
Ex 4: Given: QS  PR at S
PQ  RQ
Prove: PQS  RQS
16
Ex 5: Given: EM  AC at E
FM  BC at F
EA  FB
M is the midpoint of AB
Prove: EMA  FMB
Ex 6: Given: Q is a right angle
ST  PR at T
RT  RQ
Prove: TRS  QRS
17
Lesson 6: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
Warm up: Which pairs of triangles can be proven congruent using HL?
Are thereany other methods that will work?
a)
b)
c)
d)
Recall that when two triangles are congruent, ALL of their corresponding sides are
congruent and ALL of their corresponding angles are congruent. Thus, to prove that
two line segments are congruent or two angles are congruent….
1. Choose two triangles that contain the segments or the angles that you need to prove ≅
2. Prove the chosen triangles congruent
3. Show that the segments or angles that are to be proved congruent are corresponding
parts of congruent triangles and are therefore congruent themselves.
Ex 1: Given: 1  2 and 3  4
Prove: AD  CD
Ex 2: Given: CA  CB and AD  BD
Prove: A  B
18
Ex 3: Given: AB and CD bisect
each other at E
Prove: C  D
Ex 4:
Given: Quadrilateral ABCD, BFE , CFD ,
ADE , BE bisects CD at F, AE // BC
Prove: BF  EF
B
C
F
A
D
E
19
Ex 5:
Ex. 6:
20
Lesson 7: Exterior Angle of a Triangle
Warm up: a) Determine a method to prove the following triangles congruent.
b) Which additional sides and/or angles can then be proven as congruent using
CPCTC?
Method:______________
Method:___________________
_____________________
__________________________
_____________________
__________________________
_____________________
__________________________
_____________________
__________________________
Exterior Angle theorem: The measure of an exterior angle of a triangle is equal to the
sum of the measures of the remote interior angles.
Let’s prove this theorem:
Given: Triangle with interior angles, ∠1, ∠2, and ∠3
and exterior angle ∠4.
Prove: _____________________________________
Example 1:
a) Which of the following angles is an
exterior angle to triangle ABC?
b) If 7 is the exterior angle, name the two
remote interior angles.
c) If 12 and 8 are the two remote interior angles, what could
be the exterior angle associated with them?
21
Example 2:
Find mB
Example 3: If ∆𝐴𝐵𝐶 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠 and 𝑚∡𝐵𝐶𝐷 =
130°determine and state the 𝑚∡𝐴𝐵𝐶?
Example 4: Find mACD.
Example 5: An exterior angle at the base of
an isosceles triangle measures 110°, what is
the measure of the vertex angle?
For Examples with Quadratic Equations: REMEMBER to substitute your value(s) for x to find the
angles (even if they don’t ask for them) so that you can reject any negative angles or angles that
measure greater than 180 degrees
Example 6: Find the m  ACB
22
Lesson 8: Congruent Triangles Proofs - Isosceles Triangles
̅̅̅.
Warm up: In the figure below, determine 𝑚∡𝐶𝐵𝐷.
In the figure below, ̅̅̅̅
𝑅𝑄 ≅ ̅̅̅̅
𝑄𝑆 ≅ ̅𝑆𝑇
Determine 𝑚∡𝑃𝑄𝑅 if 𝑚∡𝑇 = 25.
Theorem: If two sides of a triangle are congruent, the angles opposite those sides are congruent.
Theorem: If two angles of a triangle are congruent, the sides opposite those angles are congruent.
*CAUTION: Both sides and both angles must be
in the SAME triangle in order to use these theorems!!!*
Ex 1:
23
Ex 2: Given: SR  ST
RM  TN
Prove: RSM  TSN
Ex. 3:
24
Ex 4:.
25
Ex 5:
26
Lesson 9: Congruent Triangles Proofs - Overlapping Triangles
Warm up: In ∆𝑃𝑄𝑅, ̅̅̅̅
𝑄𝑆 ≅ ̅̅̅̅
𝑄𝑇 and ∠𝑄𝑃𝑅 ≅ ∠𝑄𝑅𝑃
Prove ∆𝑄𝑇𝑃 ≅ ∆𝑄𝑆𝑅
Ex 1: Given: TM  TN , TR  TS
Prove: RTN  STM
Ex 2: Given: ADB , BEC
BD  BE , DA  EC
Prove: A  C
B
D
A
E
C
27
Ex 3:
Ex. 4:
Given: 1  2 , ABC  DCB
Prove: 5  6
Tips for Proofs:
1.
Mark any given information on your diagram.
2.
Look to see if the pieces you need are "parts" of the triangles that can be proven congruent.
3.
Know your definitions! If the given information contains definitions, consider these as "hints" to the solution and
be sure to use them.
4.
Stay open-minded. There may be more than one way to solve a problem
5.
Look to see if your triangles "share" parts. These common parts are automatically one set of congruent parts.
28
Lesson 10: Congruent Triangles Proofs - Two Sets of Triangles
Warm up: Name a pair of overlapping triangles in each diagram. State whether the
triangles are congruent by SSS, SAS, ASA, AAS or HL.
Ex 1:
Ex 2:
29
Ex 3:
30
Directions: Two versions of the proof are shown below.
Find the mistakes in the proofs:
̅̅̅̅ ⊥ ̅̅̅̅
̅̅̅̅ ≅ ̅̅̅̅
̅̅̅̅, 𝑷𝑸
̅̅̅̅
Given:𝑷𝑸
𝑸𝑹, ̅̅̅̅
𝑺𝑻 ⊥ 𝑻𝑼
𝑺𝑻, ̅̅̅̅
𝑷𝑺 ≅ 𝑹𝑼
̅̅̅̅||𝑺𝑻
̅̅̅̅
Prove: 𝑷𝑸
Proof 1:
Statement
Reason
̅̅̅ ⊥ ̅̅̅̅
̅̅̅, ̅̅̅̅
1. ̅̅̅̅
𝑃𝑄 ⊥ ̅̅̅̅
𝑄𝑅, ̅𝑆𝑇
𝑇𝑈, ̅̅̅̅
𝑃𝑄 ≅ ̅𝑆𝑇
𝑃𝑆 ≅ ̅̅̅̅
𝑅𝑈 1. Given
2. ∠𝑄 and ∠𝑇 are right angles
2. Perpendicular lines intersect to form right angles
̅̅̅̅ ≅ ̅̅̅̅
̅̅̅̅
3. ̅̅̅̅
𝑃𝑆 + 𝑆𝑅
𝑅𝑈 + 𝑆𝑅
3. Addition
̅̅̅̅
𝑃𝑅 ≅ ̅̅̅̅
𝑆𝑈
4. ∆𝑃𝑄𝑅 ≅ ∆𝑆𝑇𝑈
4. SSA
5. ∠𝑄𝑃𝑆 ≅ ∠𝑇𝑆𝑅
5. CPCTC
̅̅̅̅ ||𝑆𝑇
̅̅̅̅
6. 𝑃𝑄
6. If two lines are cut by a transversal so that
corresponding angles are congruent, then the lines
are parallel.
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
Proof 2:
̅̅̅ ⊥ ̅̅̅̅
Because ̅̅̅̅
𝑃𝑄 ⊥ ̅̅̅̅
𝑄𝑅 and ̅𝑆𝑇
𝑇𝑈, then ∠𝑄 and ∠𝑇 are right
angles because perpendicular lines intersect to form right
̅̅̅̅ ≅ 𝑆𝑅
̅̅̅̅ because all segments
angles. Also, we know that 𝑆𝑅
are congruent to themselves (reflexive). Therefore, since
̅̅̅̅
̅̅̅̅ ≅ ̅̅̅̅
̅̅̅̅ and ̅̅̅̅
𝑃𝑆 ≅ ̅̅̅̅
𝑅𝑈 we know that ̅̅̅̅
𝑃𝑆 + 𝑆𝑅
𝑅𝑈 + 𝑆𝑅
𝑃𝑅 ≅ ̅̅̅̅
𝑆𝑈.
Hence, we have ∆𝑃𝑄𝑅 ≅ ∆𝑆𝑇𝑈 by hypotenuse-leg. Because corresponding angles of congruent
̅̅̅̅ ||𝑆𝑇
̅̅̅̅ because if two
triangles are congruent, we can say that ∠𝑄𝑃𝑆 ≅ ∠𝑇𝑆𝑅. As a result, 𝑃𝑄
lines are cut by a transversal so that alternate interior angles are congruent, then the lines
are parallel.
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________