1. 3.2 Corresponding Parts of Congruent Triangles CPCTC: Corresponding parts of congruent triangles are congruent. Using CPCTC to prove parts of congruent triangles are congruent. Ex 1 p. 138 ▬
▬ ▬
▬ ▬ W
T Z V
Given: WZ bisects TWV
WT  WV
Prove: TZ  VZ
PROOF
Statements Reasons
___
WZ bisects TWV 1. Given
TWZ  VWZ The bisector of an angle __ ___ separates it into two  ’s
WT  WV Given
WZ  WZ Identity
Δ TWZΔ VWZ SAS
TZ TZ CPCTC
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2. Three Types of Conclusions Involving Triangles
Proving triangles congruent.
Proving corresponding pairs of congruent triangles. Note that the two triangles have to be proven congruent before you use CPCTC.
Establishing a further relationship. Note that the two triangles have to be proven congruent and also apply CPCTC first.
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3. Example 2 p. 139 __ __ Given: ZW  YX Z Y ZY  WX 2 Prove: ZY || WX 1
__ __
Given: ZW  YX
ZY  WX
Prove: ZY || WX
Plan for Proof:
Show that ΔZWX  ΔXYZ ; then we can say that 1  2 by CPCTC. Since
these are alt. int angles for ZY and WX, these lines must be parallel.
Statements Reasons
___ __ __ __
ZW  YX;ZY WX Given
ZX  ZX Identity
ΔZWX  ΔXYZ SSS
1   CPCTC
ZY || WX 5. If two lines are cut by a transversal so that the alt. Int. s are , these lines are parallel.
Z Y 2 1
W X
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4. Suggestions for Proving Triangles Congruent
Mark the figures systematically, using:
A square in the opening of each right angle.
The same number of dashes on congruent sides.
The same number of arcs on congruent angles.
Trace the triangles to be proved congruent in different colors.
If the triangles overlap, draw them separately.
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5. Right Triangles and the HL method for proving Right Triangles Congruent.
Theorem 3.2.1: If the hypotenuse and a leg of one right triangle are congruent
to the hypotenuse and a leg of a second right triangle then the triangles are congruent. (HL)
This theorem is illustrated in the
construction on p. 141, Example 3,
Fig. 3.20
Ex. 4 p. 141
Pythagorean Theorem: The square of the length (c) of the hypotenuse
of a right triangle equals the sum of the squares of the lengths (a and b)
of the legs of the right triangle. a² + b² = c²
Ex. 5 p. 142
Hypotenuse
Leg
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