1

2

Definition of Congruent TrianglesDefinition of Congruent TrianglesDefinition of Congruent TrianglesDefinition of Congruent Triangles

Δ AABBCC Δ PPQQRR

AA PP

BB QQ

CC RR

If then the corresponding sides and corresponding angles are congruent

Δ AABBCC Δ PPQQR R ,,

3

Types of Triangles classified by the sidesTypes of Triangles classified by the sidesTypes of Triangles classified by the sidesTypes of Triangles classified by the sides

1. Equilateral triangle has 3 congruent sides2. Isosceles triangle has at least 2 congruent sides3. Scalene triangle has no sides congruent

Types of Triangles classified by the anglesTypes of Triangles classified by the anglesTypes of Triangles classified by the anglesTypes of Triangles classified by the angles

1. Acute triangle has 3 acute angles. If these angles are congruent, then the triangle is also equiangular.

2. Right triangle has exactly one right angle3. Obtuse triangle has exactly one obtuse angle

AcuteAcute EquiangularEquiangular RightRight ObtuseObtuse

4

Sides of Right triangles and Isosceles triangles are given Sides of Right triangles and Isosceles triangles are given special namesspecial names

Sides of Right triangles and Isosceles triangles are given Sides of Right triangles and Isosceles triangles are given special namesspecial names

AA

CC

BB

Opposite side of < A

Adjacent sides of < A

Legs

Hypotenuse

Base Legs

Isosceles Triangle, when only 2 sides = , these sides called “legs” and the remaining side is called “base”Right Triangle, the side

opposite the right angle is called “hypotenuse”, the adjacent sides called “legs”

5

Theorem: Properties of Congruent TrianglesTheorem: Properties of Congruent TrianglesTheorem: Properties of Congruent TrianglesTheorem: Properties of Congruent Triangles

1. Every triangle is congruent to itself [ reflexive property ]

2. If then [ symmetric property ]

3. If and then

Δ AABBCC Δ PPQQRR

Δ AABBCC Δ PPQQRR

Δ PPQQRR Δ AABBCC

Δ PPQQRR Δ TTUUVV Δ AABBCC Δ TTUUVV[ transitive property ]

6

Interior and Exterior Angles of a TriangleInterior and Exterior Angles of a TriangleInterior and Exterior Angles of a TriangleInterior and Exterior Angles of a Triangle

Exterior angleExterior angle

Interior angle

Extend the anglesOriginal angle1

2 3

12 3

2 1 3

If you rotate each angle of a triangle and line them up, they add up to a straight line.

< 1 + < 2 + < 3 = line

2 3

When an auxiliary line is drawn parallel to the opposite side of the triangle, then alternate interior angles are equal.

Since 1 = 1, 2 = 2, 3 = 3, then angles of a triangle add up to a line or 180 °

7

Theorem: Triangle SumTheorem: Triangle SumThe sum of the measures of the interior angles of a triangle is The sum of the measures of the interior angles of a triangle is

180 180 °°

Theorem: Triangle SumTheorem: Triangle SumThe sum of the measures of the interior angles of a triangle is The sum of the measures of the interior angles of a triangle is

180 180 °°

PROOF: Triangles add up to 180 DIAGRAM:

GIVEN:Δ ABC

PROVE: m < 1 + m < B + < C = 180 °°

STATEMENT REASON

A line is constructed through A parallel to BC.

Parallel postulate

< B < 2 and < C < 3 Alternate Interior Angles Theorem

m < 1 + m < 2 + < 3 = 180 °° Because < 1 + < 2 + < 3 FORM A LINE

m < 1 + m < B + < C = 180 °° Substitution

2 3

32

1

B

A

C

8

Example 1: Using the Triangle Sum TheoremExample 1: Using the Triangle Sum TheoremExample 1: Using the Triangle Sum TheoremExample 1: Using the Triangle Sum Theorem

In this triangle, find m < 1, m < 2, m < 3 by using the Triangle Sum Theorem m < 3 = 180°° ─ ( 51 ° ° + 42 ° ° ) = 87 ° °

With m < 3, use Linear Pair postulate to get m < 2 = 180°° ─ 87 ° ° = 93 ° °

Again, Using the Triangle Sum Theorem m < 1 = 180°° ─ ( 28 ° ° + 93 ° ° ) = 59 ° °

132

28

42

51

Third Angles Theorem: If 2 angles of 1Third Angles Theorem: If 2 angles of 1stst triangle are congruent triangle are congruent to 2 angles of a 2to 2 angles of a 2ndnd triangle, then the 3 triangle, then the 3rdrd angles are congruent angles are congruentThird Angles Theorem: If 2 angles of 1Third Angles Theorem: If 2 angles of 1stst triangle are congruent triangle are congruent to 2 angles of a 2to 2 angles of a 2ndnd triangle, then the 3 triangle, then the 3rdrd angles are congruent angles are congruent

Theorem: Acute angles of a right triangle are complementaryTheorem: Acute angles of a right triangle are complementaryTheorem: Acute angles of a right triangle are complementaryTheorem: Acute angles of a right triangle are complementary

9

Exterior Angles Theorem: The measure of an exterior angle of Exterior Angles Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two a triangle is equal to the sum of the measures of the two

remote (non-adjacent) interior anglesremote (non-adjacent) interior angles

Exterior Angles Theorem: The measure of an exterior angle of Exterior Angles Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two a triangle is equal to the sum of the measures of the two

remote (non-adjacent) interior anglesremote (non-adjacent) interior angles

A

B

C

Exterior angle of < C = sum of the Exterior angle of < C = sum of the 2 remote interior angles < A + < B 2 remote interior angles < A + < B Exterior angle of < C = sum of the Exterior angle of < C = sum of the 2 remote interior angles < A + < B 2 remote interior angles < A + < B

10

A

B

CB

AC

A

B

Exterior Angles Theorem: Exterior Angles Theorem: As you can see by moving duplicate As you can see by moving duplicate copies of the triangle, it clearly shows how the exterior angle copies of the triangle, it clearly shows how the exterior angle

of angle C = the 2 remote angles of the triangle, of angle C = the 2 remote angles of the triangle, < A < A + + < B.< B.

Exterior Angles Theorem: Exterior Angles Theorem: As you can see by moving duplicate As you can see by moving duplicate copies of the triangle, it clearly shows how the exterior angle copies of the triangle, it clearly shows how the exterior angle

of angle C = the 2 remote angles of the triangle, of angle C = the 2 remote angles of the triangle, < A < A + + < B.< B.

11

Proving that Triangles are Congruent (same shape and size)Proving that Triangles are Congruent (same shape and size)Proving that Triangles are Congruent (same shape and size)Proving that Triangles are Congruent (same shape and size)

Side – Side – Side ( SSS ) Congruence Postulate:If 3 sides of one triangle are congruent to 3 sides of a second triangle, then the two triangles are congruent.

Side – Angle – Side ( SAS ) Congruence Postulate:If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of a second triangle, then the two triangles are congruent.

12

Proving that Triangles are Congruent (same shape and size)Proving that Triangles are Congruent (same shape and size)Proving that Triangles are Congruent (same shape and size)Proving that Triangles are Congruent (same shape and size)

Angle – Side – Angle ( ASA ) Congruence Postulate:If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of a second triangle, then the two triangles are congruent.

Angle – Angle – Side ( AAS ) Congruence Postulate:If 2 angles and a non-included side of one triangle are congruent to 2 angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

13

Examples where AAA and SSA don’t work thus Triangles are Examples where AAA and SSA don’t work thus Triangles are NOT Congruent (same shape and size)NOT Congruent (same shape and size)

Examples where AAA and SSA don’t work thus Triangles are Examples where AAA and SSA don’t work thus Triangles are NOT Congruent (same shape and size)NOT Congruent (same shape and size)

FF

DD

CC

BB

AA

As you can see, all 3 angles of both triangles are congruent to corresponding angles of other triangle, yet the triangles are NOT congruent.

As you can see, all 3 angles of both triangles are congruent to corresponding angles of other triangle, yet the triangles are NOT congruent.

AA

BB

DDCC

As you can see, triangles ABC and ABD are NOT congruent even though they have 2 congruent sides, BC BD and AB AB and a non-included angle, < A < A

As you can see, triangles ABC and ABD are NOT congruent even though they have 2 congruent sides, BC BD and AB AB and a non-included angle, < A < A

14

CPCTC: CPCTC: Corresponding Parts of Congruent Triangles are CongruentCorresponding Parts of Congruent Triangles are Congruent

CPCTC: CPCTC: Corresponding Parts of Congruent Triangles are CongruentCorresponding Parts of Congruent Triangles are Congruent

By definition, 2 triangles are congruent if and only if their corresponding parts are congruent.By definition, 2 triangles are congruent if and only if their corresponding parts are congruent.

GIVEN:A is the midpoint of MTA is the midpoint of SR

PROVE: MS ║ TR

STATEMENT REASON

1. A is the midpoint of MT 1. Given

2. MA TA 2. Definition of midpoint

3. A is the midpoint of SR 3. Given

2. SA RA 4. Definition of midpoint

5. < MAS < TAR 5. Vertical angles are congruent

6. Δ MAS Δ TAR 6. SAS Congruent Postulate

7. < M < T 7. CPCTC

8. MS ║ TR 8. Alt. Interior <‘s 2 lines are ║

AA

RR

TT

MM

SS

15

Base Angles Theorem (Isosceles Triangle)Base Angles Theorem (Isosceles Triangle)If 2 sides of a triangle are congruent,If 2 sides of a triangle are congruent,

then the angles opposite them are congruentthen the angles opposite them are congruent

Base Angles Theorem (Isosceles Triangle)Base Angles Theorem (Isosceles Triangle)If 2 sides of a triangle are congruent,If 2 sides of a triangle are congruent,

then the angles opposite them are congruentthen the angles opposite them are congruentGIVEN:NC NY

PROVE: < C < Y

STATEMENT REASON

1. Label H as the midpoint of CY 1. Ruler Postulate

2. Draw NH 2. 2 points determine a line

3. CH HY 3. Definition of midpoint

4. NH NH 4. Reflexive property of congruence

5. NC NY 5. Given

6. Δ NHC Δ NHY 6. SSS Congruence Postulate

7. < C < Y 7. CPCTC

HH

CC

YY

NN

16

Theorem: If 2 angles of a triangle are congruent, then the Theorem: If 2 angles of a triangle are congruent, then the sides opposite them are congruent.sides opposite them are congruent.

Theorem: If 2 angles of a triangle are congruent, then the Theorem: If 2 angles of a triangle are congruent, then the sides opposite them are congruent.sides opposite them are congruent.

Corollary to Base Angles theorem:If a triangle is equilateral, then it is also equiangular.

Corollary to theorem above:If a triangle is equiangular, then it is also equilateral.

Definition of a Corollary:Definition of a Corollary:A corollary is a theorem that follows easily from a theorem that has been proven.A corollary is a theorem that follows easily from a theorem that has been proven.

17

Hypotenuse – Leg (HL) Congruence TheoremHypotenuse – Leg (HL) Congruence TheoremIf the hypotenuse + a leg of a right If the hypotenuse + a leg of a right Δ are congruent to the are congruent to the

hypotenuse + leg of a 2hypotenuse + leg of a 2ndnd right right Δ, the 2 triangles are congruent , the 2 triangles are congruent

Hypotenuse – Leg (HL) Congruence TheoremHypotenuse – Leg (HL) Congruence TheoremIf the hypotenuse + a leg of a right If the hypotenuse + a leg of a right Δ are congruent to the are congruent to the

hypotenuse + leg of a 2hypotenuse + leg of a 2ndnd right right Δ, the 2 triangles are congruent , the 2 triangles are congruent

GIVEN: AC DF, CB FE., AB EG < B and < FED are right <‘s

PROVE: Δ ABC Δ DEF

STATEMENT STATEMENT

1. < B is a right angle 9. FG AC

2. < FED is a right angle 10. AC DF

3. FE DG 11. FG DF

4. < FEG is a right angle 12. < D < G

5. < B < FEG 13. < FEG < FED

6. AB EG 14. Δ FEG Δ FED

7. CB FE 15. Δ ABC Δ DEF

8. Δ ABC Δ GEF

EE

DD

GG

FFCC

BB

AA

18

19

m

π