TRIANGLE CONGRUENCE
4.1 Congruent Polygons
Polygon Congruence: Polygons are said to be congruent if all of their sides and
angles are congruent.
Naming Polygons: When naming polygons, go around the figure either
clockwise or counterclockwise listing the vertices in order.
Example 1: What are all the possible names for
the hexagon?
Corresponding Sides and Angles: The congruent sides and angles in congruent
polygons.
Example 2: Name all corresponding angles and sides if ABCD  EFGH . Then
write a congruence statement.
Polygon Congruence Postulate: Two polygons are congruent if and only if there is a
correspondence between their sides and angles such that:
 Each pair of corresponding angles is congruent.
 Each pair of corresponding sides is congruent.
Example 3: Prove that REX  FEX .
4.2 Triangle Congruence
SSS (Side-Side-Side) Postulate
If the _________________ of one triangle are congruent to the _________________ of
another triangle, then the two triangles are ___________________.
SAS (Side-Angle-Side) Postulate
If two _________________ and their _____________ _______________in one triangle
are congruent to two sides and their ______________ _______________ in another
triangle, then the two triangles are _____________________.
ASA (Angle-Side-Angle) Postulate
If two _______________ and the _______________ _______________ in one triangle
are congruent to two _________________ and the _______________ ______________
in another triangle, then the two triangles are ____________________.
Example 1: In each pair below, the triangles are congruent. Tell which triangle
congruence postulate allows you conclude so.
a.
b.
c.
Reminder: How to Write a Two-Column Proof:
 List the Given information first
 Use information from the diagram (this can also be given information).
 Give a reason for every statement
 Use given information, definitions, postulates and theorems as reasons
 List statements in order. If a statement relies on another statement, list it
later than the statement it relies on.
 End the proof with the statement you are trying to prove.
Example 2: Use the SSS Congruence Postulate
Given: AB  CB , AD  CD
Prove: ΔABD  ΔCBD
Statements
1. AB  CB
2. AD  CD
3. BD  BD
4. ΔABD  ΔCBD
Reasons
1.
2.
3.
4.
Example 3: Use the SAS Congruence Postulate
Given: RE  TE and DE  CE
Prove: ΔRED  ΔTEC
Statements
1. RE  TE
2. DE  CE
3. RED  TEC
4. ΔRED  ΔTEC
You Try! Fill in the missing reasons.
Reasons
1.
2.
3.
4.
Given: DR  AG , and RA  RG
Prove: DRA  DRG
Statements
Reasons
1.
DR  AG , RA  RG
DRA and DRG are right angles. 2.
3.
DRA  DRG
DR  DR
4.
5.
DRA  DRG
4.3 Analyzing Triangle Congruence
AAS (Angle-Angle-Side) Congruence Theorem
If two angles and a non-included side of one triangle are congruent to the corresponding
angles and the non-included side of another triangle, then the triangles are congruent.
HL (Hypotenuse-Leg) Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg
of another right triangle, then the two triangles are congruent.
Example 1: Does the diagram give enough information to show that the triangles
are congruent? If so, write a congruence statement and name the postulate or
theorem you would use.
Example 2:
Given: MNP  OPN, MPN  ONP
Prove: MNP  OPN
Statements
1. MNP  OPN
2. MPN  ONP
3. NP  NP
4. MNP  OPN
Reasons
1.
2.
3.
4.
Example 3:
Given: XQ || TR, XR bisects QT
Prove: XMQ  RMT
Statements
Reasons
1. XQ || TR
2. Q  T
3. X  R
1.
2.
3.
4. XR bisects QT
4.
5. TM  QM
6. XMQ  RMT
5.
6.
Example 4: Prove Triangles are Congruent
Given: CD  EA , AD is the perpendicular
bisector of CE
Prove: CBD  EBA
Statements
1. AD is the perpendicular bisector of CE
2. CBD and EBA are right angles
3. CBD and EBA are right triangles
4. B is midpoint of CE
5. CB  EB
6. CD  EA
7. CBD  EBA
Reasons
1.
2.
3.
4.
5.
6.
7.
Example 5: Prove Triangles are Congruent
Given: WJ  KZ , JWZ and ZKJ are
right angles
Prove: WJZ  KZJ
Statements
1. JWZ and ZKJ are right angles
2. WJZ and KZJ are right triangles
3. JZ  JZ
4. WJ  KZ
5. WJZ  KZJ
Reasons
1.
2.
3.
4.
5.
4.4 Using Triangle Congruence
In the previous lessons you learned to use SSS, SAS, ASA, AAS, and HL to prove
that two triangles are congruent. Once you know that triangles are congruent, you
can make conclusions about corresponding segments and angles because of
Corresponding Parts of Congruent Triangles are Congruent. A shorthand way of
writing this is CPCTC
Example 1:
Given: Given: AC  EC , BC  DC
Prove: A  E
Example 2:
Given: AB  CD, AE  FD, A  D
Prove: EC  FB
You Try!
Given: EOF  HOG, OFE  OGH, EG  FH
Prove: EOH is isosceles
The Isosceles Triangle Theorem:
If two sides of a triangle are congruent, then the angles opposite those sides are
congruent.
Converse of the Isosceles Triangle Theorem:
If two angles of a triangle are congruent, then the sides opposite those angles
are congruent.
Example 3:
Given: AC  BC
Prove: A  B
Plan: Show that the two triangles are congruent
by SAS, then use CPCTC
Corollary: an additional theorem that can be easily derived from the original
theorem. They are used in proofs, as definitions, postulates and theorems are.
Two Corollaries:
 The measure of each angle of an equilateral triangle is 60˚.
 The bisector of the vertex angle of an isosceles triangle is the perpendicular
bisector of the base.
4.5 Proving Quadrilateral Properties
Example 1:
Given: parallelogram PLGM with diagonal LM
Prove: LGM  MPL
Statements
1. Parallelogram PLGM has diagonal LM
2. PL|| GM
3. 3  2
4. PM || GL
5. 1  4
6. LM  LM
7. LGM  MPL
Reasons
Parallelogram Theorems:
 A diagonal of a parallelogram divides the parallelogram into two congruent
triangles.
 Opposite sides of a parallelogram are congruent.
 Opposite angles of a parallelogram are congruent.
 Consecutive angles of a parallelogram are supplementary.
 The diagonals of a parallelogram bisect each other.
Rhombus Theorems:
 A rhombus is a parallelogram.
 The diagonals of a rhombus are perpendicular.
 The diagonals are the angle bisectors.
Rectangle Theorems:
 A rectangle is a parallelogram.
 The diagonals of a rectangle are congruent.
Kite Theorems:
 The diagonals of a kite are perpendicular.
Square Theorems:
 A square is a rectangle and a rhombus.
 The diagonals of a square are congruent and are the perpendicular bisectors
of each other.
4.6 Conditions for Special Quadrilaterals
Theorems: (Use if proving a quadrilateral is a parallelogram)
 If two pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
 If one pair of opposite sides of a quadrilateral are parallel and congruent,
then the quadrilateral is a parallelogram.
 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is
a parallelogram
Theorems: (Use if proving a parallelogram is a rectangle)
 If one angle of a parallelogram is a right angle, then the parallelogram is a
rectangle
 If the diagonals of a parallelogram are congruent, then the parallelogram is a
rectangle.
Theorems: (Use if proving a parallelogram is a rhombus)
 If one pair of adjacent sides of a parallelogram are congruent, then the
parallelogram is a rhombus.
 If the diagonals of a parallelogram bisect the angles of the parallelogram,
then the parallelogram is a rhombus.
 If the diagonals of a parallelogram are perpendicular, then the parallelogram
is a rhombus.
4.7 Compass and Straightedge Constructions
 A segment congruent to a given segment
 A triangle congruent to a given triangle
 An angle bisector
 An angle congruent to a given angle
 The perpendicular bisector of a given segment
 A through a point perpendicular to a given line
4.8 Constructing Transformations
 Translating a segment
 A rotation about a point by a given angle
 A reflection across a given line
Betweenness Theorem (Converse of the Segment Addition Postulate): Given
three points P, Q, and R, if PQ + QR = PR, then P, Q, and R are collinear
and Q is between P and R.
Triangle Inequality Theorem: The sum of the lengths of any two sides of a
triangle is greater than the length of the third side.
Example: Which of the following are possible side lengths of a triangle?
a. 14, 8, 25
b. 16, 17, 23
c. 10, 8, 24