Geometry Semester 2
Homework on Basic Constructions
Name: _______________________________
You will need a straight-edge and compass for these. Show all markings.
1.
Bisect the segment:
2.
Bisect the acute angle:
3.
Bisect this obtuse angle:
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4.
Construct angle DEF to be congruent to angle ABC:
A
E
F
B
C
5.
Construct angle QPR to be congruent to angle MPQ:
Q
M
6.
P
Construct a line perpendicular to the given line at point P:
P
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7.
Construct a line perpendicular to the given line through point P:
P
8.
Use the fact that an angle inscribed in a semicircle is a right angle to construct a line
perpendicular to the given line at its endpoint P. (Hint: Choose any point above the line
to the right of P, and draw a circle with that center intersecting the line at P and another
point.)
P
9.
Construct a rectangle having these two sides:
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10.
Construct a line parallel to the given line through the given point:
11.
Locate point D below segment AB and draw ray BD so angle ABD is congruent to angle CAB:
C
A
12.
B
Trisect segment AB with points C and D. That is, locate points C and D so that AC= CD= DB:
A
B
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13.
Construct angle DAB whose measure is 3 times the measure of angle CAB:
C
A
14.
B
Angle trisection is the division of an arbitrary angle into three equal angles. It was one of the
three geometric problems of antiquity for which solutions using only compass and straightedge
were sought. The problem was proved impossible for the general case in the 1800's. Although
trisection is not possible for a general angle using only a compass and straightedge, there are
some specific angles, such as 90o that can be trisected (you can construct a 30o angle by
bisecting an angle of an equilateral triangle). Since any line segment can be trisected, you
might think the following method works:
Given angle A, mark two points, B and C on its sides equidistant from A, connect them with a
segment, trisect that segment with points P and Q, and then draw rays AP and AQ:
B
P
Q
A
C
Show that this method doesn't work by copying angle DAB you constructed in problem 13, and
then use this method to “trisect” the angle.
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