Geometry Lesson 4.2

Introduction to

Congruent Triangles

October 23, 2007

Objectives

� Students will be able to:

� Define: congruent figures

� Identify congruent figures and their corresponding sides and angles

� Find angle measures in congruent figures

� Find the measure of a 3rd angle given two congruent angles in a triangle

� Prove two triangles are congruent

Warm-Up: Review of “Congruent”

� Congruent segments have the same

_______________

� Congruent angles have the same

_______________

length

measure

1. Definition of Congruent Figures

� Geometric figures are congruentcongruent if

they have exactly the samesame sizesize and

exactly the samesame shapeshape

NOT congruentCongruent

Congruence Implies Correspondence

� Congruent figures have corresponding

anglesangles and corresponding sidessides that are

congruent

A

B

C D

E

F

∆∆∆∆ABC ≅≅≅≅ ∆∆∆∆DEF

∠∠∠∠A ≅≅≅≅ ∠∠∠∠D AB ≅≅≅≅ DE∠∠∠∠B ≅≅≅≅ ∠∠∠∠E BC ≅≅≅≅ EF∠∠∠∠C ≅≅≅≅ ∠∠∠∠F AC ≅≅≅≅ DF

Angles Sides

Get the Order Right!

� CORRESPONDING PARTS MUST MATCH!

Can also write ∆∆∆∆BCA ≅≅≅≅ ∆∆∆∆EFD, but

can NOT write ∆∆∆∆ABC ≅≅≅≅ ∆∆∆∆EFD

A

B

C D

E

F

Example 1: Naming Congruent Parts

� Write a congruence statement and name

all corresponding angles and sides

P

Q

R

Congruence Statement∆∆∆∆QRP ≅≅≅≅ ∆∆∆∆ACB

A

C

B

∠∠∠∠Q ≅≅≅≅ ∠∠∠∠A QR ≅≅≅≅ AC

∠∠∠∠R ≅≅≅≅ ∠∠∠∠C PR ≅≅≅≅ BC

∠∠∠∠P ≅≅≅≅ ∠∠∠∠B PQ ≅≅≅≅ AB

Practice 1: Naming Congruent Parts

� Write a congruence statement and name

all corresponding angles and sides

∆∆∆∆_____ ≅≅≅≅ ∆∆∆∆_____

∠∠∠∠_____ ≅≅≅≅ ∠∠∠∠_____

∠∠∠∠_____ ≅≅≅≅ ∠∠∠∠_____

∠∠∠∠_____ ≅≅≅≅ ∠∠∠∠_____

_____ ≅≅≅≅ _____

_____ ≅≅≅≅ _____

_____ ≅≅≅≅ _____

Example 2: Finding Measures

� In the figures below, NPLM ≅≅≅≅ EFGH

� Find the value of x and y

P

L

M

N72°

110°

8

10E

FG

H(7y+9)°

(2x-3)

1. What angle corresponds to ∠∠∠∠E?

2. What side corresponds to GH?

∠∠∠∠N

LM

7y + 9 = 72 y = 9

2x – 3 = 8 x = 5.5

Practice 2: Finding Measures

� In the figures, ABCDEF ≅≅≅≅ GHIJKL

� Find values for x and y

2. Third Angles Theorem� If two angles of one triangle are congruent

to two angles of another triangle, then the

third anglesthird angles are congruent

A

B

C D

E

F

If 2∠∠∠∠s ≅≅≅≅, then 3rd ∠∠∠∠s ≅≅≅≅

3rd ∠∠∠∠ 3rd ∠∠∠∠

Example 3: Third Angles Theorem

� Find the value of x

Start by writing a congruence statement:

∆∆∆∆ABC ≅≅≅≅ ∆∆∆∆DEF

Then, apply the third angles theorem:

∠∠∠∠C ≅≅≅≅ ∠∠∠∠F ∴∴∴∴m∠∠∠∠C = m∠∠∠∠F

From ∆∆∆∆ABC, m∠∠∠∠C = 40° (∆∆∆∆ sum theorem)

∴∴∴∴m∠∠∠∠D = x° = 40°

Practice 3: Third Angles Theorem� Find x & m∠∠∠∠F using the third angles theorem

A

B

C D

E

F

(2x+30)°55°

65°

3. Proving Triangles are Congruent

� Process: Use the definition of congruence

and the properties of the figure to develop

a logical argumentlogical argument

92°

92°Q

M

N

R

P

Example 4: Given the diagram, prove

∆∆∆∆PQR ≅≅≅≅ ∆∆∆∆NQM

Example 4, cont.

1. What do you need to know?

2. What do you already knowknow?

3. What can you showshow?

That all corresponding sides and angles are ≅≅≅≅

All sides are ≅≅≅≅ and m∠∠∠∠P = m∠∠∠∠N

∠∠∠∠PQR ≅≅≅≅ ∠∠∠∠NQM (vertical ∠∠∠∠s) and

∠∠∠∠R ≅≅≅≅ ∠∠∠∠M (3rd ∠∠∠∠s thm)

92°

92°Q

M

N

R

P

Example 4, cont.

Statement Reason

RP ≅≅≅≅ MN, RQ ≅≅≅≅ MQ, PQ ≅≅≅≅ NQ Given

m∠∠∠∠P = m∠∠∠∠N Given

If =, then ≅≅≅≅∠∠∠∠P ≅≅≅≅ ∠∠∠∠N

∠∠∠∠PQR ≅≅≅≅ ∠∠∠∠MQN If vertical ∠∠∠∠s, then ≅≅≅≅

∠∠∠∠R ≅≅≅≅ ∠∠∠∠M If 2∠∠∠∠s ≅≅≅≅, then 3rd ∠∠∠∠s ≅≅≅≅

∆∆∆∆PQR ≅≅≅≅ ∆∆∆∆NQM If corresp. ∠∠∠∠s and sides ≅≅≅≅, then figures ≅≅≅≅

92°

92°Q

M

N

R

P

4. Congruent Triangle Properties

� Reflexive

� Every triangle is congruent to itself

� Symmetric

� If ∆∆∆∆ABC ≅≅≅≅ ∆∆∆∆DEF, then ∆∆∆∆DEF ≅≅≅≅ ∆∆∆∆ABC

� Transitive

� If ∆∆∆∆ABC ≅≅≅≅ ∆∆∆∆DEF and ∆∆∆∆DEF ≅≅≅≅ ∆∆∆∆JKL, then

∆∆∆∆ABC ≅≅≅≅ ∆∆∆∆JKL

A

B

C D

E

F J

K

L

Closure� The design has only congruent triangles

� If the total area is 96 ft², what is the area of

one triangle

32 congruent

triangles

96 ft² / 32 = 3 ft²

per triangle

Homework Check: 4.12: hypotenuse 4: base

6: acute isosceles 8: equiangular equilateral

10: B 12: A

14: F 16: acute isosceles

18: obtuse scalene 20: obtuse isosceles

22: sometimes 24: always

26: never

32: m∠∠∠∠1 = 50°, m∠∠∠∠2 = 40°, m∠∠∠∠3 = 45°

34: m∠∠∠∠A = 33°, m∠∠∠∠B = 66°, m∠∠∠∠C = 81°, acute

36: m∠∠∠∠W = 75°, m∠∠∠∠Y = 15°, m∠∠∠∠Z = 90°, right

38: x = 10 and m∠∠∠∠ = 109° 42: all three m∠∠∠∠ = 60°

48: see next

Homework Check: 4.1 (#47)

Homework Check: 4.1 (#48)