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Use construction 2. (a) Using a working line w as one side, duplicate jA. Construct another duplicate of jA adjacent to jA, as shown. The exterior sides of the copied angles form the required angle. (b) Using a working line w as one side, duplicate jA. Construct jB adjacent to jA. Then construct jC adjacent to jB. The exterior sides of the copied angles A and C form the required angle. Note that the angle is a straight angle. (c) Using a working line w as one side, duplicate jB. Then duplicate jA from the new side of jB as shown. The difference is the required angle.
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15.3 Constructing Bisectors and Perpendiculars
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To bisect a given angle
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Given: jA (Fig. 15-5) To construct: The bisector of jA Construction: With A as center and a convenient radius, construct an arc intersecting the sides ofS at B jA S and C. With B and C as centers and equal radii, construct arcs intersecting in D. Draw AD. Then AD is the required bisector. (^ABD > ^ACD by SSS; hence, j1 > j2.)
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Fig. 15-5
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Fig. 15-6
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To construct a line perpendicular to a given line through a given point on the line
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Given: Line w and point P on w (Fig. 15-6) To construct: A perpendicular to w at P S Construction: Using construction 3, bisect the straight angle at P. Then DP is the required perpendicular; 4 DP is the required line.
CHAPTER 15 Constructions
5: To bisect a given line segment (to construct the perpendicular bisector of a given line segment)
CONSTRUCTION
Given: Line segment AB (Fig. 15-7) To construct: The perpendicular bisector of AB Construction: With A as center and a radius of more than half AB, construct arc (1). With B as center and 4 4 the same radius, construct arc (2) intersecting arc (1) at C and D. Draw CD. CD is the required perpendicular bisector of AB. (Two points each equidistant from the ends of a segment determine the perpendicular bisector of the segment.)
Fig. 15-7
CONSTRUCTION
Fig. 15-8
To construct a line perpendicular to a given line through a given external point
Given: Line w and point P outside of w (Fig. 15-8) To construct: A perpendicular to w through P Construction: With P as center and a sufficiently long radius, construct an arc intersecting w at B and C. 4 With B and C as centers and equal radii of more than half BC, construct arcs intersecting at A. Draw PA. Then 4 PA is the required perpendicular. (Points P and A are each equidistant from B and C.)
SOLVED PROBLEMS
15.3 Constructing special lines in a triangle In scalene ^ABC [Fig. 15-9(a)], construct (a) a perpendicular bisector of AB and (b) a median to AB. In ^DEF [Fig. 15-9(b)], D is an obtuse angle; construct (c) the altitude to DF and (d) the bisector of jE.
Fig. 15-9
Solutions
(a) Use construction 5 to obtain PQ the perpendicular bisector of AB. (b) Point M is the midpoint of AB. Draw CM, the median to AB. (c) Use construction 6 to obtain EG, the altitude to DF (extended). (d) Use construction 3 to bisect jE. EH is the required bisector.
CHAPTER 15 Constructions
15.4 Constructing bisectors and perpendiculars to obtain required angles (a) Construct angles measuring 90 , 45 , and 135 . (b) Given an angle with measure A (Fig. 15-10), construct an angle whose measure is 90
Fig. 15-10
Solutions
(a) In Fig. 15-10(a), mjDAB (b) In Fig. 15-10(b), mjGHJ 90 , mjCAE 90 A. 45 , mjBAE 135
15.4 Constructing a Triangle
15.4A Determining a Triangle
A triangle is determined when a set of given data fix its size and shape. Since the parts needed to prove congruent triangles fix the size and shape of the triangles, a triangle is determined when the given data consist of three sides, or two sides and the angle included by those sides, or two angles and a side included by those angles, or two angles and a side not included by those angles, or the hypotenuse and either leg of a right triangle.
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