This document defines congruence and its properties in geometry. It states that two figures are congruent if they have the same size and shape. It then defines parallel and perpendicular lines and their relationships. The document establishes several theorems about congruent angles and triangles formed by parallel lines cut by a transversal. It introduces the side-angle-side, side-side-side, and side-angle-angle postulates for triangle congruence. Finally, it proves the isosceles triangle theorem that the angles opposite congruent sides of an isosceles triangle are also congruent.

1. CONGRUENCE
Prepared by: Mrs. Monica Umali-Mora
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2. OBJECTIVES:
1. Define congruence of geometric figures
2. State the properties of congruence of geometric figures.
3. Define and state the postulates regarding parallel and
perpendicular lines
4. Establish the theorems on congruence of angles and
triangles.
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3. Congruence
Is a term widely used in the study of geometry and it is applied to figures ,
such as line segments, angles , triangles and arcs.
Note: Two geometric figures are said to be congruent if they have the
same measure. It is denoted by the symbol ≅, read as “IS CONGRUENT
TO“
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4. EXAMPLES:
A
B
K
H
AB ≅ HK if AB ≅HK
W Z
∠W ≅ ∠Z if m∠W is ≅ m∠Z
D
A
Y
P
E T
For ΔDAY ≅ΔPET, some conditions need to be satisfied
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5. CONGRUENCE is an equivalence relation. It satisfies the
following Properties.
1. REFLEXIVE PROPERTY
AB ≅AB
ㄥABC ≅ㄥABC
🛆XYZ ≅🛆XYZ
1. SYMMETRIC PROPERTY
If AB ≅CD then CD ≅ AB
If ㄥABC ≅ ㄥPQR then ㄥPQR ≅ ㄥABC
If 🛆XYZ ≅🛆QRS then 🛆QRS ≅🛆XYZ
1. TRANSITIVE PROPERTY
If AB ≅CD and CD ≅ HK then AB ≅HK
If ㄥABC ≅ ㄥPQR and ㄥPQR ≅ ㄥKLM then ㄥABC ≅ ㄥKLM
If 🛆XYZ ≅🛆QRS and 🛆QRS ≅🛆ABC then 🛆XYZ ≅🛆ABC
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6. Parallel and Perpendicular Lines
Parallel lines are lines that will never meet no matter how far they are produced
Perpendicular line are lines that form a right angles. Perpendicular bisector of a
segment is a line perpendicular to the segment at its midpoint.
A transversal is a line that intersects any two given lines.
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7. t
m
n
1 2
3 4
5 6
7 8
Line t is the transversal that intersects both line m and n.
These lines form angles with specific names.
ALTERNATE INTERIOR ANGLES- ∠3 & ∠6
∠4 &∠5
ALTERNATE EXTERIOR ANGLES- ∠2 & ∠7
∠1 &∠8
CORRESPONDING ANGLES- ∠1 & ∠5
∠2 &∠6
∠3 & ∠7
∠4 &∠8
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8. POSTULATE ON PARALLEL AND PERPENDICULAR
LINES
POSTULATE 13. PARALLEL POSTULATE
Through a point outside a given line, one and only one line can be drawn parallel
to the given line.
POSTULATE 14 .
Through a given external point, there is only one line perpendicular to a given line
POSTULATE 15.
If two line are cut by a transversal and the interior angles on the same side of the
transversal are supplementary then the two lines are parallel
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9. From what has been previously studied, let us recall the following:
1. Instances when two segments are congruent
a. The midpoint of a line segment divide the segment into two congruent
segments.
A M
B
If M is the midpoint of AB then AM ≅ MB or AM = MB
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10. From what has been previously studied, let us recall the following:
1. Instances when two segments are congruent
B. Two sides of an isosceles triangle are congruent
If ΔABC is isosceles , AB ≅ AC
A
B
C
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11. From what has been previously studied, let us recall the following:
1. Instances when two segments are congruent
C. All sides of an equilateral triangle are congruent.
If ΔABC is equilateral , XY ≅ YZ ≅ XZ
X
Y Z
D. the sides of a regular polygon are congruent 11

12. From what has been previously studied, let us recall the following:
2. Instances when two angles are congruent
If ∠1 and ∠2 are complementary, ∠3 and ∠4 are complementary , and ∠1≅∠3 ,
then ∠2≅∠4
a. All right angles are congruent
b. Complement Theorem
Complements of congruent angles are congruent.
1
2
3
4
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13. From what has been previously studied, let us recall the following:
2. Instances when two angles are congruent
If ∠1 and ∠2 are , ∠3 and ∠4 are supplementary , and ∠1≅∠3 ,
then ∠2≅∠4
C. Supplement Theorem
Supplements of congruent angles are congruent
D. Vertical Angles are congruent
1 2 3 4
1 3
2
4
∠1≅∠3 and ∠2≅∠4
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14. Theorem 1 (AIP).
If two lines are cut by a transversal and form two congruent alternate angles, then the two
lines are parallel.
t
p
q
1 3
2
Given : ㄥ1 ≅ㄥ2
Prove: p ‖ q
Proof:
Statement Reason
1. ㄥ1 ≅ㄥ2
2. ㄥ1 + ㄥ3= 180°
1. mㄥ1 = mㄥ2
1. ㄥ2 + ㄥ3= 180°
2. p ‖ q
1. Given
2. Definition of linear
pair
3. Definition of
congruence
4. Substitution
5. Postulate 15
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15. Theorem 2 (PAI).
If two lines are parallel, then the alternate interior angles formed with the transversal are
congruent.
t
p
q
1 3
2
Given : p ‖ q
Prove: ㄥ1 ≅ㄥ2
Proof: Statement Reason
1. p ‖ q
2. ㄥ3 and ㄥ2 are
supplementary
1. ㄥ3 + ㄥ2=180°
1. ㄥ3 + ㄥ1= 180°
2. mㄥ3 + mㄥ2 = mㄥ2
+ mㄥ3
3. mㄥ2 = mㄥ1
1. ㄥ1 = ㄥ2
1. Converse of postulate
15
4.
5.
6. Subtraction property of
equality
7.
Given
Substitution
Definition of linear pairs
Transitive property
Definition of congruence
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16. Theorem 2 (PCA).
In a plane, if two lines are perpendicular to the same line, then they are parallel.
Given : p ⟂ q, q ⟂ t
Prove: p ‖ q
Proof:
Statement Reason
1. p ⟂ q, q ⟂ t
1. ㄥ1 and ㄥ2 are right
angles
1. ㄥ1 ≅ ㄥ2
1. p ‖ q
1.
Given
Definition of congruence
1 2
p q
t
All right angles are
congruent
Postulate 13
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17. TRIANGLE CONGRUENCE
Two triangles are congruent if all their corresponding parts are congruent.
A
B C
X
Y X
ΔABC≅ΔXYZif AB≅XY, AC≅XZ, BC≅YZ
∠A≅∠X, ∠B≅∠Y , ∠C≅∠Z
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18. POSTULATE 16.
SAS POSTULATE
Two triangles are congruent if two pairs of corresponding sides and their included
angles are congruent.
A
B C
X
Y Z
AB≅ XY , AC ≅ XZ and ∠A ≅ ∠X
Note: any two sides may be considered, but the angle must be the on included
between these sides.
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19. POSTULATE 17.
SSS POSTULATE
Two triangles are congruent, if all the pairs of corresponding sides are congruent.
A
B C
X
Y Z
AB≅ XY , AC ≅ XZ and BC≅YZ
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20. POSTULATE 18.
SAA POSTULATE
Two triangles are congruent, if all the pairs of corresponding sides are congruent.
B
A C
Y
X Z
ΔABC≅ΔXYZif AB≅XY and ∠A ≅∠X , ∠C ≅∠Z
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21. Theorem 4. Isosceles Triangles Theorem (ITT)
If sides of a triangle are congruent , then the angles opposite them are congruent.
Given : VA ≅ VB
Prove: ㄥA ≅ㄥB
Proof:
Statement Reason
1. Draw VM from vertex
V and the midpoint of
AB
2. VM ≅ VM
1. VA ≅ VB
1. AM ≅ MB
1. ΔVAM≅ΔVM
2. ㄥA ≅ㄥB
1. Reflexive Property
1.
2. Corresponding parts of
congruent triangles are
Definition of
perpendicular line
Two sides of and
isosceles triangles are ≅
SAS CONGRUENCE
The midpoint of a lien
segment divides the
segment intotwo
congruent segments
A M B
V

22. ACTIVITY 3.4
Please answer page 283-285
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