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Proving a congruency problem
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Given: BF ' DE BF ' AC /3 /4 To Prove: AF > FC Prove: Prove nI nII
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PROOF:
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Statements 1. BF ' AC 2. /5 /6 3. BF > BF 4. BF ' DE 5. /1 is the complement of /3. /2 is the complement of /4. 6. /3 /4 7. /1 /2 8. nI nII 9. AF > FC 1. Given
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Reasons
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2. $ form rt. ; rt. are 3. Reflexive property 4. Given 5. Adjacent angles are complementary if exterior sides are to each other. 6. Given 7. Complements of are 8. ASA s 9. Corresponding parts of congruent n are
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Proving a congruency problem stated in words Prove that if the opposite sides of a quadrilateral are equal and a diagonal is drawn, equal angles are formed between the diagonal and the sides.
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CHAPTER 3 Congruent Triangles
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If the opposite sides of a quadrilateral are congruent and a diagonal is drawn, congruent angles are formed between the diagonal and the sides. Quadrilateral ABCD AB > CD, BC > AD AC is a diagonal. To Prove: /1 /4, /2 /3 Plan: Prove nI nII
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PROOF:
Given:
Statements 1. AB > CD, BC > AD 2. AC > AC 3. nI 4. /1 nII /4, /2 /3 1. Given
Reasons
2. Reflexive property 3. SSS 4. Corresponding parts of
s n are
3.2 Isosceles and Equilateral Triangles
3.2A Principles of Isosceles and Equilateral Triangles
PRINCIPLE 1:
If two sides of a triangle are congruent, the angles opposite these sides are congruent. (Base angles of an isosceles triangle are congruent.)
Thus in nABC in Fig. 3-12, if AB > BC, then /A
A proof of Principle 1 is given in 16.
Fig. 3-12
PRINCIPLE 2:
Fig. 3-13
If two angles of a triangle are congruent, the sides opposite these angles are congruent.
/C, then AB > BC.
Thus in nABC in Fig. 3-13, if /A
Principle 2 is the converse of Principle 1. A proof of Principle 2 is given in 16.
PRINCIPLE 3:
An equilateral triangle is equiangular.
/B /C.
Thus in nABC in Fig. 3-14, if AB > BC > CA, then /A
Principle 3 is a corollary of Principle 1. A corollary of a theorem is another theorem whose statement and proof follow readily from the theorem.
Fig. 3-14
CHAPTER 3 Congruent Triangles
An equiangular triangle is equilateral.
/B /C, then AB > BC > CA.
PRINCIPLE 4:
Thus in nABC in Fig. 3-15, if /A
Principle 4 is the converse of Principle 3 and a corollary of Principle 2.
Fig. 3-15
SOLVED PROBLEMS
Applying principles 1 and 3 In each part of Fig. 3-16, name the congruent angles that are opposite congruent sides of a triangle.
Fig. 3-16
Solutions
(a) Since AC > BC, /A (b) Since AB > AC, /1 /B. /2. Since BD > CD, /3 /1 /4. /D. /D /E.
(c) Since AB > AC > BC, /A (d) Since AB > BC > AC, /A
/3. Since BC > CD, /2
/ACB
/ABC. Since AE > AD > DE, /A
Applying principles 2 and 4 In each part of Fig. 3-17, name the congruent sides that are opposite congruent angles of a triangle.
Fig. 3-17
CHAPTER 3 Congruent Triangles
Solutions
(a) Since m/a (b) Since /A (c) Since /1 (d) Since /A 55 , /a /D. Hence, BC > CD. /C, BD > CD. /4 /D, CD > AD > AC. /C, BD > CD. /1, AD > BD. Since /2 /3, AB > BC. Since /2 /1
/4, AB > BD > AD. Since /2
3.10 Applying isosceles triangle principles In each of Fig. 3-18(a) and (b), nI can be proved congruent to nII. Make a diagram showing the congruent parts of both triangles and state the congruency principle involved.
Fig. 3-18
Solutions
(a) Since AB > BC, /A (b) Since AB > AC, /B /C. nI /C. nI nII by SAS [see Fig. 3-19(a)]. nII by ASA [see Fig. 3-19(b)].
Fig. 3-19
3.11 Proving an isosceles triangle problem
AB > BC AC is trisected at D and E. To Prove: /1 /2 Plan: Prove nI nII to obtain BD > BE. Given:
PROOF:
Statements 1. AC is trisected at D and E 2. AD > EC 3. AB > BC 4. /A 5. nI 7. /1 /C nII /2 1. Given
Reasons
2. To trisect is to divide into three congruent parts 3. Given 4. In a n, opposite 5. SAS
s 6. Corresponding parts of n are
sides are
6. BD > BE
7. Same as 4
CHAPTER 3 Congruent Triangles
3.12 Proving an isosceles triangle problem stated in words Prove that the bisector of the vertex angle of an isosceles triangle is a median to the base.
Solution
The bisector of the vertex angle of an isosceles triangle is a median to the base. Given: Isosceles nABC (AB > BC) BD bisects /B To Prove: BD is a median to AC Plan: Prove nI nII to obtain AD > DC.
PROOF:
Statements 1. AB > BC 2. BD bisects /B. 3. /1 5. nI /2 nII 4. BD > BD 6. AD > DC 7. BD is a median to AC. 1. Given 2. Given
Reasons
3. To bisect is to divide into two congruent parts. 4. Reflexive property. 5. SAS 6. Corresponding parts of
s n are
7. A line from a vertex of n bisecting opposite side is a median.
SUPPLEMENTARY PROBLEMS
3.1. Select the congruent triangles in (a) Fig. 3-20, (b) Fig. 3-21, and (c) Fig. 3-22, and state the congruency principle in each case. (3.1)
Fig. 3-20
Fig. 3-21
CHAPTER 3 Congruent Triangles
Fig. 3-22
In each figure below, nI can be proved congruent to nII. State the congruency principle involved.
(3.2)
State the additional parts needed to prove nI principle.
nII in the given figure by the given congruency (3.3)
Fig. 3-23
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