Download If 2 angles and a NON-INCLUDED side of 1 triangle

CHAPTER TEST REVIEW
Are YOU ready?
Amara Majeed
THEOREMS AND
POSTULATES
PROOFS
POT LUCK
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Fill in the blanks.
The _____________ Angles Theorem
states that if 2 angles of one triangle are
___________ to 2 angles of another
triangle, then the __________ pair of angles
are also ___________
The THIRD Angles Theorem states that
if 2 angles of one triangle are
CONGRUENT to 2 angles of another
triangle, then the THIRD pair of angles
are also CONGRUENT.
Match these Postulates.
1. SAS Postulate
2. SSS Postulate
3. ASA Postulate
a. If 2 angles and the included
side of one triangle are congruent
to 2 angles and the included side
of another triangle, then the
triangles are congruent.
b. If three sides of one triangle are
congruent to three sides of
another triangle, then the triangles
are congruent.
c. If 2 sides and the included
angle of one triangle are
congruent to 2 sides and the
included angle of another
triangle, then the triangles
are congruent.
1. SAS Postulate
2. SSS Postulate
3. ASA Postulate
a. If 2 angles and the included
side of one triangle are congruent
to 2 angles and the included side
of another triangle, then the
triangles are congruent.
b. If three sides of one triangle are
congruent to three sides of
another triangle, then the triangles
are congruent.
c. If 2 sides and the included
angle of one triangle are
congruent to 2 sides and the
included angle of another
triangle, then the triangles
are congruent.
Find the measure of Angle B.
B
2x+3
5x-60
15°
A
C
.D
The sum of the remote interior angles =
the exterior angle.
2x+3+15 <-----Sum of Remote interior angles
2x+18 = 5x-60 <------Set the sum of the remote interior
angles equal to the exterior angle.
18 = 3x-60 <-------Subtract 2x.
78 = 3x <--------Add 60.
x= 26<-------Divide by 3.
2(26) + 3
=
55
S
R
Given: RS if perpendicular to ST. TU is perpendicular to
ST. V is the midpoint of ST.
V
Prove: ∆RSV is congruent to ∆UTV.
T
STATEMENT
U
Reason
1) RS is perpendicular to ST, TU is perpendicular to ST, V is the
midpoint of ST.
1)
Given
2) SV is congruent to TV.
2)
Def. Of midpoint
3) Angle UVT is congruent to Angle RVS.
3)
Vertical Angles Theorem
4) RS is parallel to UT.
4)
______?_______
5) Angle S is congruent to Angle T.
5)
______?_______
6) ∆RSV is congruent to ∆UTV.
6)
______?_______
Fill in the blanks.
______________Parts of ______________ triangles are
______________.
AAS Theorem: If 2 angles and a ____________
side of 1 triangle are congruent to 2 __________
and a ____________ side of another triangle, then
the triangles are ____________.
HL Theorem: If the ___________ and a
___________ of one ___________ triangle are
congruent to the ____________ and __________
of another ___________ triangle, then the
triangles are congruent.
CORRESPONDING Parts of CONGRUENT triangles are
CONGRUENT.
AAS Theorem: If 2 angles and a NON-INCLUDED
side of 1 triangle are congruent to 2 ANGLES and
a NON-INCLUDED side of another triangle, then
the triangles are CONGRUENT.
HL Theorem: If the HYPOTENUSE and a LEG of
one RIGHT triangle are congruent to the
HYPOTENUSE and a LEG of another RIGHT
triangle, then the triangles are congruent.
S
R
Given: RS if perpendicular to ST. TU is perpendicular to
ST. V is the midpoint of ST.
V
Prove: ∆RSV is congruent to ∆UTV.
T
STATEMENT
PROVE
1) RS is perpendicular to ST, TU is perpendicular to ST, V is the
midpoint of ST.
1)
Given
2)
Def. Of midpoint
3)
Vertical Angles
theorem
4)
If 2 lines are
perpendicular
to the same
line, then they
are parallel
5)
Alternate
Interior Angles
Theorem
6)
ASA Postulate
2) SV is congruent to TV.
3) Angle UVT is congruent to Angle RVS.
4) RS is parallel to UT.
5) Angle S is congruent to Angle T.
6) ∆RSV is congruent to ∆UTV.
U
G
Given: FG is congruent to OG, GR bisects FO.
Prove: Angle F is congruent to Angle O.
F
STATEMENT
REASON
1) FG is congruent to
OG, GR bisects FO.
1) Given
2) GR is congruent to
GR.
3) _____?______
4) ∆GFR is congruent
to ∆GOR.
5) Angle F is congruent
to Angle O.
2) ____?_____
3) Def. Of Segment
Bisector
4) ___?____
5) ___?____
R
O
STATEMENT
REASON
1) FG is congruent to
OG, GR bisects FO.
1) Given
2) GR is congruent to
GR.
3) FR is congruent to
RO.
4) ∆GFR is congruent
to ∆GOR.
5) Angle F is
congruent to Angle
O.
2) Reflexive POC
3) Def. Of
Segment
Bisector
4) SSS Postulate
5) CPCTC
Which theorem or postulate can be
used to prove these two triangles
congruent?
HL THEOREM
This is a right triangle.
Therefore, since it is
supplementary to this
angle, both triangles
are classified as right
triangles.
This line segment is congruent to
itself. It is a leg of both triangles, since
it makes a right angle with the other
leg. Therefore, since the hypotenuse
and leg of one triangle is congruent to
the hypotenuse and leg of the other
triangle, according to the HL theorem,
the two triangles are congruent.
BONUS ROUND!
Name the 2 simple corollaries present
in this unit. ($500 each)
NAME THIS POSTULATE: The sum
of the lengths of any 2 sides of a
triangle is greater than the length of
the third side. ($500)
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The acute angles of a right triangle are
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Teammeasure of each angle of an equiangular triangle
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This is the TRIANGLE INEQUALITY POSTULATE.
T
Given: E is the midpoint of MJ. TE is perpendicular to
MJ.
Prove: ∆MET is congruent to ∆JET.
M
STATEMENT
REASON
1)
E is the midpoint of MJ,
TE is perpendicular to
MJ.
1)
Given
2)
Reflexive POC
2)
TE is congruent to TE.
3)
____________
3)
ME is congruent to EJ.
4)
____________
4)
Angle TEM=90°, Angle
TEJ=90°
5)
____________
6)
____________
5)
Angle TEM is congruent
to Angle TEJ.
6)
∆MET is congruent to
∆JET.
E
J
STATEMENT
REASON
1) E is the midpoint of
MJ, TE is
perpendicular to
MJ.
1) Given
2) TE is congruent to
TE.
4) Def. Of
perpendicular
lines
3) ME is congruent to
EJ.
4) Angle TEM=90°,
Angle TEJ=90°
5) Angle TEM is
congruent to Angle
TEJ.
6) ∆MET is congruent
to ∆JET.
2) Reflexive POC
3) Def. of midpoint
5) Transitive POC
6) SAS postulate
P
Given: Isosceles ∆PQR, base QR, PA is congruent
to PB
A
Prove: AR is congruent to BQ
Q
STATEMENT
REASON
Isosceles ∆PQR, base QR, PA is
congruent to PB
1)
Given
2)
Reflexive POC
2)
Angle P is congruent to Angle P
3)
_________________
3)
PQ is congruent to PR
4)
_________________
4)
∆QPB is congruent to ∆RPA
5)
_________________
5)
AR is congruent to BQ
1)
B
R
STATEMENT
REASON
1) Isosceles ∆PQR,
base QR, PA is
congruent to PB
1) Given
2) Angle P is
congruent to Angle
P
2) Reflexive POC
3) Def. Of Isosceles ∆
3) PQ is congruent to
PR
4) ∆QPB is congruent
to ∆RPA
4) SAS postulate
5) AR is congruent to
BQ
5) CPCTC
Can a triangle have the following
measures?
1)
2, 4, 4
2)
8, 8, 8
3)
3, 1, 1
4)
5, 6, 7
1)YES
2)YES
3)NO
4)YES
Any questions answered incorrectly will cost
DOUBLE the amount of points. (Each question is
200 points.)
State which postulate or theorem, if any, proves that
the two triangles are congruent.
1)
2)
1) NONE
2) AAS THEOREM