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CHAPTER 3 Congruent Triangles
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(a) In Fig. 3-23(a) by SSS. (b) In Fig. 3-23(a) by SAS. (c) In Fig. 3-23(b) by ASA. (d) In Fig. 3-23(b) by SAS. (e) In Fig. 3-23(c) by SSS. (f) In Fig. 3-23(c) by SAS.
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In each part of Fig. 3-24, the congruent parts needed to prove nI that are congruent.
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nII are marked. Name the remaining parts (3.4)
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Fig. 3-24
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In each part of Fig. 3-25, find x and y.
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(3.5)
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Fig. 3-25
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Prove each of the following. (a) In Fig. 3-26: Given: BD ' AC D is midpoint of AC. To Prove: AB > BC BD is altitude to AC. BD bisects /B. To Prove: /A /C (c) In Fig. 3-27: Given: / 1 > / 2, BF > DE BF bisects /B. DE bisects /D. /B and /D are rt. . To Prove: AB > CD BC > AD E is midpoint of BC. F is midpoint of AD. AB > CD, BF > DE To Prove: /A /C
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(3.6)
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(b) In Fig. 3-26: Given:
(d) In Fig. 3-27: Given:
Fig. 3-26
Fig. 3-27
CHAPTER 3 Congruent Triangles
(g) In Fig. 3-29:
(e) In Fig. 3-28:
/1 /2 CE bisects BF. To Prove: /C /E Given:
Given: CD > C D , AD > A D CD is altitude to AB. C D is altitude to A B . To Prove: /A /Ar Given: CD bisects /C. C D bisects /Cr /C /Cr, /B /Br, BC B C . To Prove: CD > C D
( f ) In Fig. 3-28: Given: BF and CE bisect each other. To Prove: BC > EF
(h) In Fig. 3-29:
Fig. 3-28 Fig. 3-29
Prove each of the following:
(3.7)
(a) If a line bisects an angle of a triangle and is perpendicular to the opposite side, then it bisects that side. (b) If the diagonals of a quadrilateral bisect each other, then its opposite sides are congruent. (c) If the base and a leg of one isosceles triangle are congruent to the base and a leg of another isosceles triangle, then their vertex angles are congruent. (d) Lines drawn from a point on the perpendicular bisector of a given line to the ends of the given line are congruent. (e) If the legs of one right triangle are congruent respectively to the legs of another, their hypotenuses are congruent. 3.8. In each part of Fig. 3-30, name the congruent angles that are opposite sides of a triangle. (3.8)
Fig. 3-30
CHAPTER 3 Congruent Triangles
In each part of Fig. 3-31, name the congruent sides that are opposite congruent angles of a triangle. (3.9)
Fig. 3-31
In each part of Fig. 3-32, two triangles are to be proved congruent. Make a diagram showing the congruent parts of both triangles and state the reason for congruency. (3.10)
Fig. 3-32
In each part of Fig. 3-33, nI, nII, and nIII can be proved congruent. Make a diagram showing the congruent parts and state the reason for congruency. (3.10)
Fig. 3-33
Prove each of the following: (a) In Fig. 3-34: Given: AB > AC F is midpoint of BC. /1 /2 To Prove: (b) In Fig. 3-34: Given: FD > FE AB > AC AD > AE FD ' AB, FE ' AC To Prove: BF > FC (c) In Fig. 3-35: Given: AB > AC /A is trisected. AD > AE AB > AC DB > BC CE > BC To Prove: AD > AE
(3.11)
To Prove: (d) In Fig. 3-35: Given:
CHAPTER 3 Congruent Triangles
Fig. 3-34
Fig. 3-35
Prove each of the following: (a) The median to the base of an isosceles triangle bisects the vertex angle.
(3.12)
(b) If the bisector of an angle of a triangle is also an altitude to the opposite side, then the other two sides of the triangle are congruent. (c) If a median to a side of a triangle is also an altitude to that side, then the triangle is isosceles. (d) In an isosceles triangle, the medians to the legs are congruent. (e) In an isosceles triangle, the bisectors of the base angles are congruent.
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