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Congruent Triangles
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3.1 Congruent Triangles
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Congruent figures are figures that have the same size and the same shape; they are the exact duplicates of each other. Such figures can be moved on top of one another so that their corresponding parts line up exactly. For example, two circles having the same radius are congruent circles. Congruent triangles are triangles that have the same size and the same shape. If two triangles are congruent, their corresponding sides and angles must be congruent. Thus, congruent triangles ABC and ArBrCr in Fig. 3-1 have congruent corresponding sides (AB ArCr , BC BrCr , and /Cr). AC ArCr ) and congruent corresponding angles (/A /Ar, /B /Br, and /C
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Fig. 3-1
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(Read nABC nArBrCr as Triangle ABC is congruent to triangle A-prime, B-prime, C-prime. ) Note in the congruent triangles how corresponding equal parts may be located. Corresponding sides lie opposite congruent angles, and corresponding angles lie opposite congruent sides.
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3.1A Basic Principles of Congruent Triangles
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PRINCIPLE 1:
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If two triangles are congruent, then their corresponding parts are congruent. (Corresponding parts of congruent triangles are congruent.)
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nArBrCr in Fig. 3-2, then /A /Ar, /B /Br, /C /Cr, a ar, b br, and c cr.
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Thus if nABC
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Fig. 3-2
CHAPTER 3 Congruent Triangles
Methods of Proving that Triangles are Congruent
PRINCIPLE 2:
(Side-Angle-Side, SAS) If two sides and the included angle of one triangle are congruent to the corresponding parts of another, then the triangles are congruent.
br, c cr, and /A /Ar in Fig. 3-3, then nABC nArBrCr.
Thus if b
Fig. 3-3
PRINCIPLE 3:
(Angle-Side-Angle, ASA) If two angles and the included side of one triangle are congruent to the corresponding parts of another, then the triangles are congruent.
/Ar, /C /Cr, and b br in Fig. 3-4, then nABC nArBrCr.
Thus if /A
Fig. 3-4
PRINCIPLE 4:
(Side-Side-Side, SSS) If three sides of one triangle are congruent to three sides of another, then the triangles are congruent.
ar, b br, and c cr in Fig. 3-5, then nABC nArBrCr.
Thus if a
Fig. 3-5
SOLVED PROBLEMS
Selecting congruent triangles From each set of three triangles in Fig. 3-6 select the congruent triangles and state the congruency principle that is involved.
CHAPTER 3 Congruent Triangles
Solutions
(a) nI (b) nII (c) nI nII, by SAS. In nIII, the right angle is not between 3 and 4. nIII, by ASA. In nI, side 10 is not between 70 and 30 . nII nIII by SSS.
Fig. 3-6
Determining the reason for congruency of triangles In each part of Fig. 3-7, nI can be proved congruent of nII. Make a diagram showing the equal parts of both triangles and state the congruency principle that is involved.
Fig. 3-7
Solutions
s (a) AC is a common side of both n [Fig. 3-8(a)]. nI
nII by ASA. nII by SAS. nII by SSS.
(b) /1 and /2 are vertical angles [Fig. 3-8(b)]. nI
s (c) BD is a common side of both n [Fig. 3-8(c)]. nI
Fig. 3-8
Finding parts needed to prove triangles congruent State the additional parts needed to prove nI nII in the given figure by the given congruency principle.
CHAPTER 3 Congruent Triangles
Fig. 3-9
(a) In Fig. 3-9(a) by SSS. (b) In Fig. 3-9(a) by SAS. (c) In Fig. 3-9(b) by ASA. (d) In Fig. 3-9(c) by ASA. (e) In Fig. 3-9(c) by SAS.
Solutions
(a) If AD > BC, then nI (b) If /1 /4, then nI nII by SSS. nII by SAS. nII by ASA. nII by ASA. nII by SAS.
(c) If BC > CE, then nI (d) If /2 /3, then nI
(e) If AB > DE, then nI
Selecting corresponding parts of congruent triangles In each part of Fig. 3-10, the equal parts needed to prove nI parts that are congruent.
nII are marked. List the remaining
Fig. 3-10
Solutions
Congruent corresponding sides lie opposite congruent angles. Congruent corresponding angles lie opposite congruent sides. (a) Opposite 45 , AC > DE, Opposite 80 , BC > DF. Opposite the side of length 12; /C (b) Opposite AB and CD, /3 /A /C. /3. Opposite BE and EC, /1 /4. Opposite /5 and /6, AB > CD. (c) Opposite AE and ED, /2 /2. Opposite BC and AD, /1 /D.
/4. Opposite common side BD,
CHAPTER 3 Congruent Triangles
Applying algebra to congruent triangles In each part of Fig. 3-11, find x and y.
Fig. 3-11
Solutions
(a) Since nI nII, by SSS, corresponding angles are congruent. Hence, 2x or y 20. (b) Since nI and y 5 24 or x 20 12, and 3y 26 or x 6, 60
nII, by SSS, corresponding angles are congruent. Hence, x 42 or y 47.
(c) Since nI nII, by ASA, corresponding sides are congruent. Then 2x 3y 8 and x 2y. Substituting 2y for x in the first of these equations, we obtain 2(2y) 3y + 8 or y 8. Then x 2y 16.
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