Geometry Notes TC 1: Side - Angle - Side Congruent Polygons

[Pages:3]Geometry Notes TC ? 1: Side - Angle - Side Congruent Polygons Review: Two polygons are congruent if Also, two polygons are congruent if (and only if)

1. All pairs of corresponding angles are congruent AND 2. All pairs of corresponding sides are congruent.

Ex: If ABC PQR, then

a. All pairs or corresponding parts are congruent

A P, B Q, C R, AB PQ , BC QR , AC PR

b. There is a rigid motion for which the image of ABC is PQR.

1. Translate along AP

2. Rotate CCW around P until PB' coincides with PQ B

3. Reflect over PQ

C' C B'

A

P C"

R Q

Problem: Saying two figures are congruent if one is the image of the other under a rigid motion is a good definition of congruence. But it is not always a convenient method to prove two figures are congruent.

Ex: Are the triangles below congruent?

If they are, then the transformation

1. Translate along AD 2. Rotate CCW around D until

AB' coincides with DE will map ABC onto DEF.

C A

B

B'

C'

F

A

E

But how can we be sure that the triangles actually map perfectly one onto the other?

By itself, the transformation DOES NOT PROVE THAT

C F

A D

B E

Proving Two Triangles Congruent

If all three pairs of corresponding sides are congruent and all three pairs of corresponding angles are congruent, then two triangles must be congruent.

Is it possible to prove two triangles congruent without proving all six pairs of corresponding parts congruent? If so, what is the least number of congruent pairs of corresponding parts we need?

One pair of sides?

One pair of angles?

Two pairs of sides?

Two pairs of angles?

One pair of each sides, angles?

Side-Angle-Side

If two sides and the included angle of one triangle are all congruent to the corresponding sides and angle of a second triangle, then the two triangles are congruent. Abbreviation: SAS

Given: ABC and A'B'C'

B'

AB A' B ' , AC A'C ' , and A A'

Note: A is called the included angle for sides

AB and AC because it is the angle formed by

C'

A'

those two sides (where those two sides meet).

B C

Show via rigid motions that A'B'C' ABC.

A 1. Translate along A'A

A' A, B' B", C' C"

B'

2. Rotate CW around A until ray AB" coincides with ray AB .

C''' B

Because AB A'B', B" B.

C'

Also, C" C''' .

B" A'

C

3. Reflect over AB .

Because BAC BAC''', ray AC''' will coincide with ray AC . Also, b/c AC'''

C"

A

AC , C''' C.

Therefore, the image of A'B'C' is ABC so the two triangles are congruent.

Ex: Given: AB CD , AB || CD Prove: ABC CDA

A

B

D

C

Statement 1. AB CD (S) 2. AB|| CD 3. BAC DCA (A) 4. AC AC (S) 5. ABC CDA

Reason 1. Given 2. Given 3. When lines are parallel, alt. int. s are 4. Reflexive 5. SAS (1, 3, 4)

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