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

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. 2.

Ex: If ABC PQR, then a. All pairs or corresponding parts are congruent

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

C

B

A

P 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

B

C

F

will map ABC onto DEF.

A

D

E

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

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.

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

B' C'

C'''

B" A'

B C

C"

A

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

A

B

D

C

Geometry HW: Triangle Congruence ? 1 Side Angle Side

For the following four problems, determine if the information given in the diagram is sufficient to prove the triangles congruent and give a reason for your answer.

1. F B

D 2. D

C 3. B

4. D

C

A

C

E

E

A

B

A

C

D A

B

In the next three problems, name the pair of corresponding sides or angles that would need to be proved

congruent, in addition to the ones already shown, in order to prove the triangles are congruent by SAS.

C

5. A

D

6.

7. A

C

B

E

A

D

B

B F

C E D

Write complete geometry proofs for the following (diagrams below): B

8. Given: AB AD , AC bisects BAD Prove: ABC ADC

A

C

D

9. Given: AS RT , A is the midpoint of RT Prove: RAS TAS

R

A

S

T

10. Given: PQ RS , PQ || RS , QUTS , QU ST Prove: PQT RSU

P

T S

Q U R

C

11. a. In a triangle, what is 1) an altitude? 2) a median? 3) an angle bisector?

A

B

b. Copy the diagram at right onto your own paper and on it draw and label altitude CP , median CM and

angle bisector CX .

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