Geometry - 4.3Congruent Triangles

Congruent, Corresponding Angles/Sides

A P

B Q

C R

AB PQ

BC QR

CA RP

ABC PQR

Two figures are congruent when their corresponding sides and corresponding angles are congruent.

Corresponding Angles

Corresponding Sides

There is more than one way to write a congruence statement, but the you must list the corresponding angles in the same order.

Naming Congruent Parts

ABC ZXY

A Z

B X

C Y

XY BC

YZ AC

XZ AB

Write a congruence statement for the triangles below. Identify all pairs of congruent parts.

Corresponding Angles Corresponding Sides

Identify Corresponding Congruent Parts

Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.

Answer: All corresponding parts of the two polygons are congruent. Therefore, ABCDE RTPSQ.

Sides:

Angles:

Third Angle Thm

A D B E C F

Third Angle Thm. - If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

If and then,

Properties of Congruent Triangles

ABC ABC

,If ABC DEF then DEF ABC

, ,If ABC DEF and DEF HIJ

then ABC HIJ

Transitive Property of Congruent Triangles

Reflexive Property of Congruent Triangles

Symmetric Property of Congruent Triangles

Proof of Third Angle Thm

• 1)• 2) m<A = m<D, m<B = m<E• 3) m<A + m<B + m<C = 180• 4) m<D + m<E + m<F = 180• 5) m<A + m<B + m<C =

m<D + m<E + m<F • 6) m<C = m<F • 7)

• 1) Given• 2) Def of congruent angles• 3) Triangle Sum Thm• 4) Triangle Sum Thm• 5) Substitution

• 6) Subtraction• 7) Def of congruent angles

A D B E and

A D B E

C F

C F

Given:

Prove:

Using the Third Angle Thm.

22 87 180

109 180

71

m A

m A

m A

4 15 71

4 56

14

m D m A

x

x

x

Find the value of x.

Determining Triangle Congruency

EFG HJG

Decide whether the triangles are congruent. Justify your reasoning.

From the diagram all corresponding sides are congruent and that <F and <H are congruent.

<EGF and <HGJ are congruent because of Vertical angles.

<E and <J are congruent because of the third angle theorem

Since all of the corresponding sides and angles are congruent,

Using Properties of Congruent Figures

ABCD KJHL 4 3 9

4 12

3

x

x

x

5 12 113

5 125

25

y

y

y

In the diagram,

a) Find the value of x.

b) Find the value of y.

Use Corresponding Parts of Congruent Triangles

In the diagram, ΔITP ΔNGO. Find the values of x and y.

O P

6y – 14 = 406y = 54

y = 9

x – 2y = 7.5

x – 2(9) = 7.5

x – 18 = 7.5

x = 25.5

Answer: x = 25.5, y = 9

A. x = 4.5, y = 2.75

B. x = 2.75, y = 4.5

C. x = 1.8, y = 19

D. x = 4.5, y = 5.5

In the diagram, ΔFHJ ΔHFG. Find the values of x and y.

2. LNM PNO 2. Vertical Angles Theorem

Proof:

Statements Reasons

3. M O

3. Third Angles Theorem

4. ΔLMN ΔPON

4. Def of Congruent Triangles

1. Given1.

Prove: ΔLMN ΔPON

Proving Two Triangles Congruent

• 1) O is the midpt of MQ and PN

• 2)• 3)• 4)• 5)

• 1) Given

• 2) Alt. Int. <‘s Thm.• 3) Vertical <‘s• 4) Def of Midpoint• 5) Def of Congruent Tri<‘s

, ||MN QP MN PQ

,MO QO PO NO

,OMN OQP MNO QPO MON QOP

MNO QPO

Given:

O is the midpt of MQ and PN

Prove:

, ||MN QP MN PQ

MNO QPO

Practice Problems

•Pg.257 #8,9-15(odds),19-23(odds),24

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