Math 106 - Cooley
Math For Elementary Teachers II
OCC
Activity #25 – Congruent Triangles
California State Content Standard – Measurement and Geometry for Grade Seven
3.0 Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by
constructing figures that meet given conditions and by identifying attributes of figures:
3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence
means about the relationships between the sides and angles of the two figures.
Definition – Congruent Triangles
Two triangles are congruent if, and only if, there is a correspondence of vertices of the triangles such that
the corresponding sides and corresponding angles are congruent.
J
A
B
K
C
L
So, the following are all true:
ABC  LKJ , A  L , B  K , C  J , AB  LK , CB  JK , AC  LJ
Definition – Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
If two triangles are congruent, then all corresponding sides and all corresponding angles are also congruent.
Property – Side–Side–Side (SSS)
If the three sides of one triangle are respectively congruent to the three sides of another triangle, then the
two triangles are congruent.
Property – Triangle Inequality
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Property – Side–Angle–Side (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 two triangles are congruent.
Theorem – Isosceles Triangle Theorem
The angles opposite the congruent sides of an isosceles triangle are congruent.
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Math 106 - Cooley
Math For Elementary Teachers II
OCC
Activity #25 – Congruent Triangles
Theorem – Thales’ Theorem
Any triangle ABC inscribed in a semicircle with diameter AB has a right angle at point C.
C
A
O
B
Property – Angle–Side–Angle (ASA)
If two angles and the included side of one triangle are congruent to the two angles and the included side of
another triangle, then the two triangles are congruent.
Theorem – Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent; that is, the triangle is
isosceles.
Property – Angle–Angle–Side (AAS)
If two angles and a non-included side of one triangle are respectively congruent to two angles and the
corresponding non-included side of a second triangle, then the two triangles are congruent.
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Math 106 - Cooley
Math For Elementary Teachers II
OCC
Activity #25 – Congruent Triangles
 Exercises: The two triangles shown are congruent.
1)
The corresponding vertices are: ___________________________________________________.
2)
The corresponding angles are: ___________________________________________________.
3)
The corresponding sides are: ___________________________________________________.
4)
CYW  _________.
 Exercises: Each question below shows two triangles, with arcs and tick marks identifying congruent
parts. If it is possible to conclude that the triangles are congruent, describe what property or
theorem you used. If you cannot be sure that the triangles are congruent, state, “No
conclusion possible.”
5)
6)
7)
8)
9)
10)
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Math 106 - Cooley
Math For Elementary Teachers II
OCC
Activity #25 – Congruent Triangles
 Exercises: Each question below shows two triangles, with given information about the triangles. Fill in
the statements and reasons and prove that the two triangles are congruent, if possible.
11)
Given: 1  2 , A  E , AC  EC
Prove: ABC  EDC
STATEMENTS
REASONS
1)
1)
2)
2)
3)
3)
4)
4)
In traditional geometry, the three given statements in the previous example are usually written in the first
line of the statements. Thus, the previous example could have been answered using only two steps.
12)
Given: AB  CB , AD  CD
Prove: ABD  CBD
STATEMENTS
REASONS
1)
1)
2)
2)
3)
3)
4)
4)
-4-
Math 106 - Cooley
Math For Elementary Teachers II
OCC
Activity #25 – Congruent Triangles
 Exercises: Each question below shows two triangles, with given information about the triangles. Fill in
the statements and reasons and prove that the two triangles are congruent, if possible.
13)
Given: Quadrilateral PRST with PR  ST , PRT  STR
Prove: PRT  STR
STATEMENTS
REASONS
1)
1)
2)
2)
3)
3)
4)
4)
14)
Given: MO  QP , M  Q
Prove: MOP  QPO
STATEMENTS
REASONS
1)
1)
2)
2)
3)
3)
4)
4)
-5-
Math 106 - Cooley
Math For Elementary Teachers II
OCC
Activity #25 – Congruent Triangles
 Exercises: Each question below shows two triangles, with given information about the triangles. Fill in
the statements and reasons and prove that the two triangles are congruent, if possible.
15)
Given: PQR  PSR , QPR  SPR
Prove: PQR  PSR
STATEMENTS
REASONS
1)
1)
2)
2)
3)
3)
4)
4)
16)
Given: M  T , L  S
Prove: MPL  TPS
STATEMENTS
REASONS
1)
1)
2)
2)
3)
3)
4)
4)
-6-
Math 106 - Cooley
Math For Elementary Teachers II
OCC
Activity #25 – Congruent Triangles
 Exercises: Each question below shows two triangles, with given information about the triangles. Fill in
the statements and reasons and prove that the two triangles are congruent, if possible.
17)
Given: X is the midpoint of WY , WZ  YZ
Prove: WXZ  YXZ
STATEMENTS
REASONS
1)
1)
2)
2)
3)
3)
4)
4)
5)
5)
18)
Given: EF || HI , G is the midpoint of EI
Prove: EFG  IHG
STATEMENTS
REASONS
1)
1)
2)
2)
3)
3)
4)
4)
5)
5)
6)
6)
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