review : solving systems
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Review : Solving Systems. x+y. x. 2y+3. 12. Find the values of x and y that make the following triangles congruent. Congruent Triangles (CPCTC). Two triangles are congruent triangles if and only if the c orresponding p arts of those c ongruent t riangles are c ongruent. - PowerPoint PPT PresentationTRANSCRIPT
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Review: Solving Systems
x 2y+3x+y
12
Find the values of x and y that make the following triangles congruent.
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Congruent Triangles (CPCTC)
Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent.
• Corresponding sides are congruent
• Corresponding angles are congruent
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Congruence Statement
When naming two congruent triangles, order is very important.
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Third Angle Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
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Congruence Shortcuts
Side-Angle-Side (SAS) Congruence Postulate:
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
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Side-Side-Side Congruence Postulate
SSS Congruence Postulate:If the three sides of one triangle are congruent to
the three sides of another triangle, then the two triangles are congruent.
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Congruence Shortcuts
Angle-Side-Angle (ASA) Congruence Postulate:
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
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Congruence Shortcuts
Angle-Angle-Side (AAS) Congruence Theorem:If two angles and a non-included side of one
triangle are congruent to the corresponding two angles and the non-included side of another triangle, then the two triangles are congruent.
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Base Angles Theorem:
If two sides of a triangle are congruent, then the angles opposite them are congruent.
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Converse of the Base Angles Theorem:
If two angles of a triangle are congruent, then the sides opposite them are congruent.
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Equilateral Triangle Theorem
A triangle is equilateral if and only if it is equiangular.
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Practice
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Practice
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Practice
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Congruence in Right Triangles
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Vocabulary Right Triangles
Leg
Leg
HypotenuseA
CB
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Hypotenuse-Leg Theorem
Hypotenuse-Leg (HL) Congruence Theorem:
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.
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• To use the HL Theorem, you must show that three conditions are met:
• There are two right triangles
• The triangles have congruent hypotenuses
• There is one pair of congruent legs
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Using the HL Theorem
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Using the HL Theorem
Statements Reasons
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
, ofbisector theis CEAD
of Defn.
s rt. are & EBACBD
EBCB
EACD
Thm. HL
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Using the HL Theorem
Statements Reasons
1.
2.
3.
4.
1.
2.
3.
4.
srt are and RPQPRS
srt of Defn.
of Prop. Refl.
QRSP
RPQPRS
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Which are congruent by HL?
3in3in
5in
5in
5in
3in
J
HG
FE
D
A
CB
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Which are congruent by HL?
3in3in
5in
5in
5in
3in
J
HG
FE
D
A
CB
HFJ DEG
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Prove the triangles are congruent
R
QP
S
4. SRP QPR
3. PR PR
2. QPR SRP
1. QPR and SRPare right, SP RQ
Given
All Right angles are congruent
Reflexive
HL Theorem
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What else do you need to prove the triangles are congruent?
V T
R
XIs RT XT? orIs XV TV?
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Prove the two triangles are congruent
T is the midpoint of RV
V
R
U
TS
3. RST TUV
1. T is the midpoint of RVS and U are right anglesRS TU
2. RT TV
1. Given
2. Definition of midpoint
3. HL Theorem
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