Geometry Section 2.2 Notes

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Transcript Geometry Section 2.2 Notes

CHAPTER
2
2.2 Proofs Involving
Congruence
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Slide 4-1
Congruent Segments
Two segments that have the same length are called
congruent segments. The symbol ≅ means
congruent.
We mark congruent segments in a figure with
exactly the same number of tick marks.
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Slide 4-2
Definition
Two angles that have the same measure are called
congruent angles. Recall that the symbol ≅ means
congruent.
We mark congruent angles with exactly the same
number of arcs, as shown in the figures
below.
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Slide 4-3
Definitions
Congruent figures have the exact same shape and
size. All the figures below are congruent. As shown
below, a flip, rotate (turn), or slide does not affect
whether figures are congruent because they still
have the same shape and size.
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Slide 4-4
Definitions
When figures are congruent, their corresponding
sides are congruent, and their corresponding angles
are congruent.
Corresponding Angles
& Sides
AB  RS
A   R
B   S
C   T
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BC  ST
CA  TR
Slide 4-5
Example
Naming Congruent Parts
For the two figures, we
are given that ABCD ≅ TRQS.
Solution
The figures help, but having the congruent angles
listed in the corresponding order is all that is
needed. ABCD ≅ TRQS
Angles: A  T , B  R, C  Q, D  S
Sides: AB  TR, BC  RQ, CD  QS , DA  ST
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Slide 4-6
Example
Proving Triangles Are Congruent
Given: segments LM ≅ LO, MN ≅ ON,
∠M ≅ ∠O, ∠MLN ≅ ∠OLN
Prove: ΔLMN ≅ ΔLON
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Slide 4-7
Example
Proving Triangles Are Congruent
Statements
Reasons
1. LM  LO, MN  ON
1. Given
2. LN  LN
2. Reflexive Property of Congruence
3. M  O ,
MLN  OLN
3. Given
4. MNL  ONL
4. Third Angles Theorem
5. LMN  LON
5. Definition of congruent
triangles
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Slide 4-8
Postulate 4.3-2 Side-Angle-Side (SAS)
Postulate
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Slide 4-9
Example
Proving Triangles Are Congruent
Given: The figure with congruent segments shown
by equal number of tick marks
Prove: ABE  CBD
Statements
Reasons
1. AB  BC, EB  BD
1. Given
2. 1  2
2. Vertical Angles Theorem
3. ABE  CBD
3. SAS Postulate
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Slide 4-10
Postulate 4.4-1 Angle-Side-Angle (ASA)
Postulate
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Slide 4-11
Example
Identifying ASA
Multiple Choice: Choose two triangles that are
congruent by the ASA Postulate. Explain why.
a.
b.
c.
d.
Solution
Choices b and d are congruent by ASA because for
these two triangles, the sides marked congruent are
the included sides of the two congruent angles.
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Slide 4-12
Postulate 4.3-1 Side-Side-Side (SSS)
Postulate
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Slide 4-13
Example
Proving Triangles Are Congruent
Given: AB  CD, AC  BD
Prove: ABC  DCB
Statements
Reasons
1. AB  CD, AC  BD
1. Given
2. CB  CB
2. Reflexive Property of Congruence
3. ABC  DCB
3. SSS Postulate
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Slide 4-14