4.3 congruent triangles

4.3 Congruent Triangles

We will……name and label corresponding parts

of congruent triangles.…identify congruence transformations.

Corresponding parts Corresponding parts of congruent trianglesof congruent triangles

Triangles that are the same size and shape are congruent triangles.

Each triangle has three angles and three sides. If all six corresponding parts are congruent, then the triangles are congruent.

Corresponding parts of congruent triangles

A

C

B

X

Z

Y

If ΔABC is congruent to ΔXYZ , then vertices of the two triangles correspond in the same order as the letter naming the triangles.

ΔABC = ΔXYZ~

Corresponding parts of congruent triangles

A

C

B

X

Z

Y

This correspondence of vertices can be used to name the corresponding congruent sides and angles of the two triangles.

ΔABC = ΔXYZ~

Definition of Definition of Congruent Triangles Congruent Triangles

(CPCTC)(CPCTC)

Two triangles are congruent if Two triangles are congruent if and only if their corresponding and only if their corresponding parts parts are congruent.are congruent.

CPCTCCPCTCCCorresponding orresponding PParts of arts of CCongruent ongruent TTriangles are riangles are

CCongruentongruent

Answer: Since corresponding parts of congruent triangles are congruent,

ARCHITECTURE A tower roof is composed of congruent triangles all converging toward a point at the top. Name the corresponding congruent angles and sides of HIJ and LIK.

Answer:

The support beams on the fence form congruent triangles.

b. Name the congruent triangles.

a. Name the corresponding congruent angles and sides of ABC and DEF.

Answer: ABC DEF

Properties of Properties of Triangle CongruenceTriangle CongruenceCongruence of triangles is reflexive, symmetric, and transitive.

REFLEXIVEREFLEXIVEK

J

L

K

J

LΔJKL = ΔJKL~~


SYMMETRICSYMMETRICK

J

L

Q

P

R

If If ΔΔJKL = JKL = ΔΔPQR,PQR,

then then ΔΔPQR =PQR = ΔΔJKL.JKL.

~~

~~


TRANSITIVETRANSITIVEK

J

L

Q

P

R

If If ΔΔJKL = JKL = ΔΔPQR, andPQR, and

ΔΔPQR = PQR = ΔΔXYZ, thenXYZ, then

ΔΔJKL =JKL = ΔΔXYZ.XYZ.

~~

~~

~~

Y

X

Z

IDENTIFY CONGRUENCE IDENTIFY CONGRUENCE TRANSFORMATIONSTRANSFORMATIONS

B

A

C

B

A

C

If you slide If you slide ΔΔABC down and to the ABC down and to the right, it is still congruent to right, it is still congruent to ΔΔDEF.DEF.

E

D

F


B

A

C B

A

C

If you turn If you turn ΔΔABC, ABC,

it is still congruent to it is still congruent to ΔΔDEF.DEF.

E

D

F


B

A

C

B

A

C

If you flip If you flip ΔΔABC, ABC,

it is still congruent to it is still congruent to ΔΔDEF.DEF.

E

D

F

COORDINATE GEOMETRY The vertices of RST are R(─3, 0), S(0, 5), and T(1, 1). The vertices of RST are R(3, 0), S(0, ─5), and T(─1, ─1). Verify that RST RST.

Use the Distance Formula to find the length of each side of the triangles.

Use a protractor to measure the angles of the triangles. You will find that the measures are the same.

Answer: The lengths of the corresponding sides of two triangles are equal. Therefore, by the definition of congruence,

In conclusion, because ,

COORDINATE GEOMETRY The vertices of RST are R(─3, 0), S(0, 5), and T(1, 1). The vertices of RST are R(3, 0), S(0, ─5), and T(─1, ─1). Name the congruence transformation for RST and RST.

Answer: RST is a turn of RST.

COORDINATE GEOMETRY The vertices of ABC are A(–5, 5), B(0, 3), and C(–4, 1). The vertices of ABC are A(5, –5), B(0, –3), and C(4, –1).

Answer:

Use a protractor to verify that corresponding angles are congruent.

a. Verify that ABC ABC.

Answer: turn

b. Name the congruence transformation for ABC and ABC.

BOOKWORK:

p. 195 #9 – 19,

#22 – 25 (just name the congruence

transformation)

HOMEWORK:

p.198 Practice Quiz

4.3 congruent triangles

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