Transcript Section 21.1

1. Given right triangle ABC with angle measures as
indicated in the figure. Find x and y.
§ 21.1
Angles x and z will be complementary to 43 and y will be equal.
2. It is given that ABC is equilateral and that DE  BC.
Prove that ADF is isosceles.
FCE will be a 30-60-90 triangle making  AFD = 30 since it is a
vertical angle.  BAF is an exterior angle of AFD and is equal to
60. The exterior angle is equal to the sum of the two opposite interior
angles hence  D + 30 = 60 or  D = 30 and sides opposite equal
angles have equal measure and thus ADF is isosceles.
3. Using “Sketchpad” investigate the phenomenon that follows from the following
construction.
Construct segment AB, its midpoint C, and the perpendicular bisector of AB.
Locate point D on the perpendicular. What kind of triangle is ABD?
Construct ray AD and segment BD.
Locate point E on ray AD so that A – D – E, then select points E, D, and B,
and construct the angle bisector DF of  EDB using Angle Bisector under
CONSTRUCTION.
Now drag points B and D and observe the effect on the figure. What do you notice?
Have you discovered a theorem? Try to write a proof for it.
Slope j = 0.00
8
Slope l = 0.00
6
E
D
4
l
F
2
A
-10
j
C
B
-5
5
-2
-4
-6
-8
Bisector of exterior angle is parallel to third side.
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4. Prove your choice of Corollary A, B, or C of Theorem 2.
Corollary A. If two lines are cut by a
transversal, then the two lines are parallel iff
a pair of interior angles on the same side of
the transversal are supplementary.
2 1
1 2
IF: Given lines parallel. Prove supplementary
If lines are parallel then the angles marked 2 are congruent by Theorem 2. But
angle 1 and 2 form a straight line and are supplementary. QED.
ONLY IF: Given supplementary angles lines parallel. Prove lines parallel.
If angles marked 1 and 2 in blue are supplementary and the red 1 and blue 2 are
supplementary show red 1 and blue 1 are congruent. But the red 1 and blue 1
are congruent shows that alternate interior angles are congruent and thus by
theorem 1 the lines are parallel.
4. Prove your choice of Corollary A, B, or C of Theorem 2.
Corollary B. If two lines are cut by a
transversal, then the two lines are parallel iff
a pair of corresponding angles are
congruent.
1
1 2
1
IF: Given lines parallel. Prove corresponding angles congruent.
Corollary A proved that blue 1 and blue 2 are supplementary. And blue 2 and
red 1 are supplementary since they form a straight line. Hence by transitive
property blue 1 and red 1 are congruent. QED.
ONLY IF: Given corresponding angles congruent. Prove lines parallel.
Given that the red 1 and blue 1 are congruent. The red 1 and green 1 are
congruent by vertical angles hence the blue and green 1’s are congruent by
transitive property. By theorem 1 the lines are parallel.
4. Prove your choice of Corollary A, B, or C of Theorem 2.
Corollary C. If two lines are cut by a
transversal, then the two lines are parallel iff
a pair of alternate interior angles are
congruent.
Both parts follow directly from theorem1 and theorem 2.
2 1
1 2
5. Prove that parallel lines are everywhere equidistant.
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A
3
B
l
1
2
C
4
D
m
Given l and m parallel. Construct AC and BD perpendicular to m.
 1 =  2 and  3 =  4 by the Z corollary. AD = AD giving ABD = DCA
by ASA. And AC = BD by CPCTE.
6. Using “Sketchpad” construct a triangle and the angle bisector of an internal and external angle of that
triangle at a vertex (Use the following procedure if you are uncertain of how to use Sketchpad”.).
i. Construct ABC using the segment tool.
ii. Construct ray AC and locate D on that ray so that A – C - D.
iii. Using Angle Bisector under CONSTRUCTION, select points A, C, B to construct the
bisector CE of ACB. Repeat this for the bisector CF of DCB.
a. Drag point A, keeping it on (or parallel) to a fixed line through B. What happens to FCE?
Does it change position and measure?
b. What seems to be true of ABC when ray CF is parallel to AB?
c. Could you prove your observations?
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mECF = 90.00
Slope AB = 3.00
Slope j = 3.01
m CA = 9.42 cm
m BC = 9.41 cm
6
B
4
F
E
j
2
-10
-5
5
C
A
-2
-4
10
D
7. Given ABC with D on side AB and AD = DB = CD. Prove ACB = 90.
C
1
1
A
1 + 1 + 2 + 2 = 180 so 1 + 2 = 90
2
2
D
B
8. Consider all taxi hyperbolas. Find a relationship between the hyperbola and the
difference of the distances, PA  PB.
PA - PB is the length between the curves on a line between the foci.