Download Math 53 Winter N09 3.2 Corresponding Parts of Congruent

Math 53 "Winter ’09"
3.2 "Corresponding Parts of Congruent Triangles"
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Objectives:
*
Using CPCTC to establish congruency in triangles
*
How to plan a proof
*
Properties of right triangles
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Key Concepts:
CPCTC
Hypotenuse and Legs of a Right Triangle
HL
Pythagorean Theorem
Square Roots Property
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Preliminaries:
Recall that the de…nition of congruent triangles states that all six parts of one triangle are congruent
respectively to the six corresponding parts of the second triangle.
Example 1: (Congruent triangles)
Given:
R
6
6
Prove:
and
6
V are right
6
s
1=6 2
RST =
V ST
CPCTC:
kCorresponding parts of congruent triangles are congruentk
Page: 1
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
Note:
3.2
For triangles that have been proved congruent, CPCTC may be used to establish that either
two lines segments or two angles are congruent.
Example 2: (Using CPCTC)
Given:
Prove:
HJ ? KL and
HK = HL
KJ = JL
Example 3: (Planning a proof)
Given:
Prove:
ZW = Y X and ZY = W X
ZY W X
Because many proofs depend on establishing congruent triangles, we need to consider the following suggestions
Suggestions for a proof that involves congruent triangles
1: Mark the …gures systematically, using the same number of dashes on congruent sides,
the same number of arcs on congruent angles, and a square in each right angle.
2: Trace the triangles to be proved congruent in di¤erent colors.
3: If the triangles overlap, draw them separately.
Page: 2
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
3.2
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)
Example 4: (Using Theorem 3.2.1)
Given:
6
1 and
6
2 are right
6
s
H is the midpoint of F K
F G HJ
Prove:
F G = HJ
Pythagorean Theorem:
The square of the length of the hypotenuse of a right triangle equals the sum of squares
of the lengths of the legs of the right triangle
Square Roots Property
Let x represent the length of a line segment, and let p represent a positive number.
If x2 = p; then
:
Page: 3
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
3.2
Example 5: (Using the Pythagorean Theorem)
Given:
Find:
ABC is a right triangle, a = 4; and
b=3
c
Example 6:
Given:
DB ? BC
and
CE ? DE
AB = AE
Prove:
BDC =
ECD
Page: 4
Bibiana Lopez