Congruent Triangles

Two geometric figures are
congruent if, and only if,
one figure overlaps
another figure completely

Which of the following figures is congruent to the yellow triangle?
D
E

B
D
E

C
D
E

The symbol ≅ represents the
congruence of two figures.
A B
C D
𝐴𝐵 ≅ 𝐶𝐷
The alphabet is written in the ORDER of the
points and the pair of overlapping sizes.

Two line segments are congruent if,
and only if, they have the same length.
Two angles are congruent if, and only
if, they have the same measure.

Triangles are congruent when their
corresponding sides are equal in length
and their corresponding angles are equal
in size.
Side-Side-Side (SSS)
Congruence Theorem

A
BC
M
NO
𝐴𝐵 ≅ 𝑀𝑁 𝐵𝐶 ≅ 𝑁𝑂 𝐶𝐴 ≅ 𝑂𝑀
∆𝐴𝐵𝐶 ≅ ∆𝑀𝑁𝑂

A B
C
D E
F
30° 30°
AB ≅ DE AC ≅ EF CAB ≅ FDE
D E
F
30°
∆ABC ≅ ∆DEF

PQ ≅ XY
P Q
R
30°
X Y
Z
30°
PQ ≅ XY RPQ ≅ ZXY
X Y
Z
30°
∆PQR ≅ ∆XYZ

A B
C
30°
P Q
R
30°
AB ≅ PQ AC ≅ PRCAB ≅ RPQ
P Q
R
30°
∆ABC ≅ ∆PQR

If two sides and the included angle of one
triangle are congruent to two sides and
included angle of another triangle, then
the two triangles are congruent.
Side-Angle-Side (SAS)
Congruence Theorem

1) Show that triangles ABC and PQR are congruent.
A B
C
2 cm
4 cm
30°
P Q
R
2 cm
4 cm
30°
Statement Reason
1) AB ≅ PQ 1) Given
2) BC ≅ QR 2) Given
3) B ≅ Q 3) Given
4) ∆ABC ≅ ∆PQR 4) SAS

2) Show that triangles ABC and CDA are congruent given that BC ≅ DA and BC // DA.
B
D
C
A
Statement Reason
1) BC ≅ DA 1) Given
2) BC // DA 2) Given
3) BCA ≅ DCA 3) Alternate interior angles are congruent
4) AC ≅ CA 4) Common side/ Reflexive Property
5) ∆ABC ≅ ∆CDA 5) SAS

3) Given that DR ⊥ AG and AR ≅ GR, prove that ∆DRA ≅ ∆DRG.
D
A GR
Statement Reason
1) AR ≅ GR 1) Given
2) DR ⊥ AG 2) Given
3) DRA and DRG are right angles 3) Definition of perpendicular lines
4) DRA ≅ DRG 4) Right angles are congruent
5) DR ≅ DR 5) Common side/ Reflexive Property
6) ∆DRA ≅ ∆DRG 6) SAS

4) Given that point C is the midpoint of BF, and AC ≅ CE, prove that ∆ABC ≅ ∆EFC.
Statement Reason
1) AC ≅ EC 1) Given
2) C is the midpoint of BF 2) Given
3) BC ≅ FC 3) Definition of a midpoint
4) BCA ≅ FCE 4) Vertical angles are congruent
5) ∆ABC ≅ ∆EFC 5) SAS
A
B
C
E
F

5) Given that BD bisects reflex CDA, and CD ≅ AD, prove that ∆BCD ≅ ∆BAD.
A
B D
C
Statement Reason
1) CD ≅ AD 1) Given
2) BD bisects reflex CDA 2) Given
3) BDC ≅ BDA 3) Definition of angle bisector
4) BD ≅ BD 4) Common side/ Reflexive Property
5) ∆BCD ≅ ∆BAD 5) SAS

A B
C
5 cm
55° 30°
P Q
R
5 cm
55° 30°

A B
C
5 cm
55° 30°
P Q
R
5 cm
55° 30°
If two angles and the included side of one triangle
are congruent to two angles and the included side
of another triangle, then the two triangles are
congruent.
Angle-Side-Angle (ASA)
Congruence Theorem

1) From the given figure, ABO ≅ OCD and BO ≅ OC. Prove that AB ≅ CD.
Statement Reason
1) ABO ≅ OCD 1) Given
2) BO ≅ OC 2) Given
3) AOB ≅ DOC 3) Vertical angles are congruent
4) ∆AOB ≅ ∆DOC 4) ASA
5) AB ≅ CD 5) Congruent Parts of Congruent Triangles
are Congruent (CPCTC)
A
B
C
D
O

2) From the given figure, show that ∆ABC ≅ ∆ABD.
A B
C D
Statement Reason
1) BAD ≅ ABC 1) Given
2) CAD ≅ DBC 2) Given
3) BAD + CAD ≅ ABC + DBC 3) Addition Property
4) BAD + CAD ≅ BAC 4) Partition Postulate
5) ABC + DBC ≅ ABD 5) Partition Postulate
6) BAC ≅ ABD 6) Substitution Property
7) AB ≅ BA 7) Common side/ Reflexive Property
8) ∆ABC ≅ ∆ABD 8) ASA

3) From the given figure, show that ∆AFD ≅ ∆BCE.
A B
F E D C
G
Statement Reason
1) AFD ≅ BCE ≅ GDE ≅ GED 1) Given
2) EF ≅ DC 2) Given
3) ED ≅ DE 3) Reflexive Property
4) EF + ED ≅ DC + DE 4) Addition Property
5) EF + ED ≅ FD 5) Partition Postulate
6) DC + DE ≅ CE 6) Partition Postulate
7) FD ≅ CE 7) Substitution Property
8) ∆AFD ≅ ∆BCE 8) ASA

4) Given AP ≅ AQ, CP ⊥ AB and BQ ⊥ AC, prove that BQ ≅ CP.
Statement Reason
1) AP ≅ AQ 1) Given
2) CP ⊥ AB and BQ ⊥ AC 2) Given
3) APC and AQB are right angles 3) Definition of Perpendicular lines
4) APC ≅ AQB 4) Right angles are congruent
5) BAQ ≅ CAP 5) Common angle
A
P
B
Q
C
6) ∆BAQ ≅ ∆CAP 6) ASA
7) BQ ≅ CP 7) CPCTC

5) Given EF ≅ BC, AB // ED and C ≅ F, prove that AF ≅ CD.
A
B
C
D
E
F
Statement Reason
1) C ≅ F 1) Given
2) AB // ED 2) Given
3) ABF ≅ DEC 3) Alternate interior angles are congruent
4) EF ≅ BC 4) Given
5) EB ≅ BE 5) Reflexive Property
6) EF + EB ≅ BC + BE 6) Addition Property
7) EF + EB ≅ BF 7) Partition Postulate
8) BC + BE ≅ EC 8) Partition Postulate
9) BF ≅ EC 9) Substitution Property
10) ∆ABF ≅ ∆DEC 10) ASA
11) AF ≅ CD 11) CPCTC

A
B C
1) B ≅ E
2) C ≅ F
3) BC ≅ EF
4) ∆ABC ≅ ∆DEF ASA
D
E F
5) A ≅ D CPCTC

A
B C
D
E F
If two angles and the non-included side of one
triangle are congruent to two angles and the
corresponding non-included side of another
triangle, then the two triangles are congruent.
Angle-Angle-Side (AAS)
Congruence Theorem

1) Given that ACB ≅ DCB and BAC ≅ BDC, prove that ∆ABC ≅ ∆DBC
A
D
CB
Statement Reason
1) ACB ≅ DCB 1) Given
2) BAC ≅ BDC 2) Given
3) BC ≅ BC 3) Common side
4) ∆ABC ≅ ∆DBC 4) AAS

2) Given that ∆ABC is an isosceles triangle and AD ⊥ BC at point D, prove that
∆ABD ≅ ∆ACD.
A
B CD
Statement Reason
1) ∆ABC is an isosceles triangle 1) Given
2) ABD ≅ ACD 2) Definition of Isosceles triangle
3) AD ⊥ BC 3) Given
4) ADB and ADC are right angles 4) Definition of perpendicular lines
5) ADB ≅ ADC 5) Right angles are congruent
6) AD ≅ AD 6) Common side
7) ∆ABD ≅ ∆ACD 7) AAS

A
B
C
D
E
3) Given that C is the midpoint of BE and AB//DE, prove that ∆ABC ≅ ∆DEC
Statement Reason
1) C is the midpoint of BE 1) Given
2) BC ≅ EC 2) Definition of a midpoint
3) AB//DE 3) Given
4) ABC ≅ DEC 4) Alternate interior angles are congruent
5) BAC ≅ EDC 5) Alternate interior angles are congruent
6) ∆ABC ≅ ∆DEC 6) AAS

4) Given that ABC ≅ DEF, AC ≅ FD and BC//EF, prove that ∆ABC ≅ ∆DEF
B E
A D C F
Statement Reason
1) AC ≅ FD 1) Given
2) ABC ≅ DEF 2) Given
3) BC//EF 3) Given
4) ACB ≅ DFE 4) Corresponding angles are congruent
5) ∆ABC ≅ ∆DEF 5) AAS

Congruent Triangles

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