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