congruent triangles indolent ichabod two triangles are congruent if and only if all of their...

CONGRUENT TRIANGLES

Indolent Ichabod

Two triangles are congruent IF AND ONLY IF

all of their corresponding parts are congruent

Indolent Ichabod

Ichabod was indolent (that means lazy).

His job: make sure trusses are identical so roofs don’t collapse

Your job: help him save time for a nap!

Indolent Ichabod

Find:

SHORTCUTS for measuring parts of a triangle to guarantee congruence

Corresponding Parts

Parts of a figure that are “in the same spot”

Translation: easy

Rotation or reflection: A bit tougher

Congruent Figures

Same size and shape.

Have congruent corresponding parts.

OPTION 1Pick up and overlap (or measure all parts!)

How many parts (hint: sides & angles) would you have to measure on a triangle?

OPTION 2 Measure only SOME

parts…

Indolent Ichabod likes that idea…

Proving figures are Congruent

1. Will a pair of triangles with one Side congruent always be congruent to one another?

2. List all combinations of parts that can be measured…

Congruence SHORTCUTS Measure only:

Shortcut 1: S

Shortcut 2: A

Shortcut 3: AA

Shortcut 4: SS

Congruence SHORTCUTS Measure only:

Shortcut 5: SA

Shortcut 6: AAA

Shortcut 7: SSS “Side-Side-Side”

Shortcut 8: SAS “Side-Angle-Side”

Congruence SHORTCUTS Measure only:

Shortcut 9: ASA “Angle-Side-Angle”

Shortcut 10: AAS “Angle-Angle-Side”

Shortcut 11: SSA

Shortcut 12: HL “Hypotenuse-Leg”(for right triangles only)

Test the SHORTCUTS

Test the shortcuts with Congruent Triangle Manipulatives SSSSASAASSSAASA

CONGRUENT TRIANGLE MANIPULATIVES

Which ones work?

SSS

SAS

ASA

AAS

Improved Ichabod

What’s this mean for ICHABOD?

He must measure at least 3 parts of two different trusses to make sure they’re identical (BETTER than doing all 6!)

Measuring 3 parts doesn’t always do it!(SSA)

USING the SHORTCUTSIchabod has to determine if he could use the next

few pairs of triangular trusses or if he should slap a rejected sticker on them and send ‘em back to the manufacturer.

FOR EACH PAIR:

What should he do?

Which conjecture proves it?

USING the SHORTCUTS

4 ft.

5 ft.

6 ft.

6 ft.

5 ft.

4 ft.

USING the SHORTCUTS

10 ft.100º 7 ft.

7 ft. 10 ft.100 º

USING the SHORTCUTS

92º

92º

43º

53º

8 ft.

8 ft.

USING the SHORTCUTS

4 feet4 feet

4 feet4 feet

70

7070

70

USING the SHORTCUTS

8 feet

6 feet

100

8 feet

6 feet

100

SSSGiven: BD bisects ACAB = CB

D

B

A C

Which triangles are congruent?

Which “shortcut” tells you so?

SASGiven: BD is a perpendicular bisector of AC

D

B

A C

Which triangles are congruent?

Which “shortcut” tells you so?

Given: EF = HF and FG = FI

Which triangles are congruent?

Which “shortcut” tells you so?

F

E

H

G

I

What’s this mean for ICHABOD?

IF Ichabod’s trusses were right triangles, he could get away with measuring the hypotenuse and a leg only.

Which conjecture would he really be using?

Why does it work in this case?