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CONGRUENT TRIANGLES
Indolent Ichabod
Two triangles are congruent IF AND ONLY IF
all of their corresponding parts are congruent
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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!
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Indolent Ichabod
Find:
SHORTCUTS for measuring parts of a triangle to guarantee congruence
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Corresponding Parts
Parts of a figure that are “in the same spot”
Translation: easy
Rotation or reflection: A bit tougher
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Congruent Figures
Same size and shape.
Have congruent corresponding parts.
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OPTION 1Pick up and overlap (or measure all parts!)
How many parts (hint: sides & angles) would you have to measure on a triangle?
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OPTION 2 Measure only SOME
parts…
Indolent Ichabod likes that idea…
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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…
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Congruence SHORTCUTS Measure only:
Shortcut 1: S
Shortcut 2: A
Shortcut 3: AA
Shortcut 4: SS
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Congruence SHORTCUTS Measure only:
Shortcut 5: SA
Shortcut 6: AAA
Shortcut 7: SSS “Side-Side-Side”
Shortcut 8: SAS “Side-Angle-Side”
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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)
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Test the SHORTCUTS
Test the shortcuts with Congruent Triangle Manipulatives SSSSASAASSSAASA
CONGRUENT TRIANGLE MANIPULATIVES
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Which ones work?
SSS
SAS
ASA
AAS
Improved Ichabod
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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)
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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?
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USING the SHORTCUTS
4 ft.
5 ft.
6 ft.
6 ft.
5 ft.
4 ft.
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USING the SHORTCUTS
10 ft.100º 7 ft.
7 ft. 10 ft.100 º
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USING the SHORTCUTS
92º
92º
43º
53º
8 ft.
8 ft.
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USING the SHORTCUTS
4 feet4 feet
4 feet4 feet
70
7070
70
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USING the SHORTCUTS
8 feet
6 feet
100
8 feet
6 feet
100
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SSSGiven: BD bisects ACAB = CB
D
B
A C
Which triangles are congruent?
Which “shortcut” tells you so?
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SASGiven: BD is a perpendicular bisector of AC
D
B
A C
Which triangles are congruent?
Which “shortcut” tells you so?
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Given: EF = HF and FG = FI
Which triangles are congruent?
Which “shortcut” tells you so?
F
E
H
G
I
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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?