4.2 Congruence and Triangles

Geometry - 4.2Congruence &

Triangles

Congruent, Corresponding Angles/Sides

A P

B Q

C R

AB PQ

BC QR

CA RP

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.

ΔABC ≅ ΔPQR

Naming Congruent Parts

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

ΔABC ≅ ΔZXY

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 Theorem. - 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

Transitive Property of Congruent Triangles

Reflexive Property of Congruent Triangles

Symmetric Property of Congruent Triangles

ΔABC ≅ ΔABC

If ΔABC ≅ ΔDEF, then ΔDEF ≅ ΔABC

If ΔABC ≅ ΔDEF and ΔGHI ≅ ΔDEF, then ΔABC ≅ ΔGHI

Proof of Third Angle ThmGiven: <A ≅ <D, <B ≅ <E

Prove: <C ≅ <F

1. <A ≅ <D, <B ≅ <E 1. Given

2. m<A = m<D, m<B = m<E 2. Def’n of Congruent Angles

3. m<A + m<B + m<C = 180 3. Triangle Sum Theorem

4. m<D + m<E + m<F = 180 4. Triangle Sum theorem5. m<A + m<B + m<C = m<D + m<E + m<F 5. Transitive Property

6. m<D + m<E + m<C = m<D + m<E + m<F 6. Substitution Property

7. m<C = m<F 7. Subtraction Property

8. <C ≅ <F 8. Def’n of Congruent Angles

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

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,

ΔEFG ≅ ΔHJG

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 midpoint 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

Given:

O is the midpt of MQ and PN

Prove:

, ||MN QP MN PQ

ΔMNO ≅ ΔQPO

ΔMNO ≅ ΔQPO

Practice Problems

•Textbook p206: 14-32 even, 35