Transcript If two angles and the include side of one triangle

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TO REVIEW, SELECT FROM THE FOLLOWING OPTIONS:
SAS
Congruent
figures
AAS
HL
DONE!
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examples!
WHICH FIGURES ARE CONGRUENT?
Yes! Since we have parallel
lines, <F = <J and <G = <K by
alternate interior angles. Then
<FHG = <KHJ because they’re
vertical angles. So now we
know all pairs of
corresponding parts are
congruent!!!
Yes! <LNM = < PNQ because
they’re vertical angles. Then
<M = <P by the third angles
theorem.
So now
we know
Yes! Because
of the
third all
pairs theorem
of corresponding
parts
angles
<JHG = <JHI.
congruent!!!
Also, JHare
= JH
by reflexive. So
now we know all pairs of
corresponding parts are
congruent!!!
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We know <Q = <T because it was given
We know <QSR = <TSV because they’re vertical angles
We know <R = <V by the third angle theorem
But, we have no information about the sides of the
triangles, and there is no theorem or postulate proving
triangles are congruent without knowing something
about the sides.
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Example:
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examples!
WHICH TRIANGLES ARE CONGRUENT BY SSS?
YES! All three pairs
of corresponding
sides are congruent!
YES! All three pairs
of corresponding
sides are congruent
because KO = KO!
YES! All three pairs
of corresponding
sides are congruent
because YQ = YQ!
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•
•
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We know that ZW = ZX because it was given
We know that <W = <X because it was given
We know ZY = ZY by the reflexive property
But, we know nothing about the third sides (WY
and XY), so we can’t use SSS here
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Example:
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examples!
*remember that EF = EF
WHICH TRIANGLES ARE CONGRUENT BY SAS?
Yes! <WXZ = <XZY because
they’re alternate interior
angles and XZ = XZ by
reflexive.
Yes! We have two pairs of
congruent corresponding
sides, and the included
angle is marked right. All Yes! <PTF =
<STG because
right angles are congruent
they’re vertical
by Theorem 2.4
angles.
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• We know that <ONR = <NRE because it was given
• We know NR = NR by the reflexive property
• But, we know nothing about any of the other sides,
and we need to know that NO = RE to use SAS
because the sides need to include the angle
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Example:
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examples!
WHICH TRIANGLES ARE CONGRUENT BY ASA?
Yes! IJ = IJ by reflexive.
Yes!Yes!
<JLK
<MLPtwo pairs
We=have
because
ofthey’re
congruent
vertical
angles. angles,
corresponding
and their included
sides are also marked
congruent.
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• We know that <JML is a right angle
• Since our book allows us to assume that all things
that appear to be lines, are lines, we also know
<JMK is right since <JMK and <JML are
supplementary (they form a linear pair)
• We know JM = JM by the reflexive property
• But, we know nothing about any of the other
angles, and we need to know that <MJL = <JMK to
use ASA because the angles need to include the
congruent side
Example:
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examples!
WHICH TRIANGLES ARE CONGRUENT BY AAS?
Yes! GF = GF by
reflexive.
Yes! <LMO =
<NMP because
they’re vertical
angles.
Yes! <HED =
<GEF because
they’re vertical
angles.
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• We know that <ABC = <CDE because all right
angles are congruent (Theorem 2.4)
• We know <BCA = <ECD because they are vertical
angles
• But, we know nothing about any of the sides, and
we need to know that either AC = CE or that AB =
DE to use AAS
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Example:
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examples!
WHICH TRIANGLES ARE CONGRUENT BY HL?
Yes! PR = PR by
reflexive.
Yes! BD = BD by
reflexive.
Yes! We have two right
triangles, GJ = JI, and
GH = IH, which is all
we need of HL.
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• In the left triangle, we have a leg marked as 7
units, and a hypotenuse marked as 25 units
• In the right triangle, the sides marked 7 and 25
units are both legs
• We can’t use HL unless there are two hypotenuses
marked congruent, and a pairs of corresponding
legs marked congruent
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• First of all, it spells a bad word, and
your parents would be disappointed
in your teachers
• Second, if you had two triangles
with a pair of congruent angles not
included by two pairs of congruent
sides, this would not be enough
information to prove the triangles
are congruent (believe me, many
intelligent mathematicians have
tried!)
• It has actually been proven that ASS
will never work on any triangle