1. Given right triangle ABC with angle measures as indicated in the figure. Find x and y.

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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.

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l

j

Slope l = 0.00

Slope j = 0.00

CA B

D E

F

Bisector of exterior angle is parallel to third side.

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.

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.

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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.

ONLY IF: Given supplementary angles lines parallel. Prove lines parallel.

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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.

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.

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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.

ONLY IF: Given corresponding angles congruent. Prove lines parallel.

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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.

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5. Prove that parallel lines are everywhere equidistant.

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lBA

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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j

m BC = 9.41 cmm CA = 9.42 cm

Slope j = 3.01Slope AB = 3.00

mECF = 90.00

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7. Given ABC with D on side AB and AD = DB = CD. Prove ACB = 90.

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1 + 1 + 2 + 2 = 180 so 1 + 2 = 90

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.