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

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

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

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  1. § 21.1 • 1. Given right triangle ABC with angle measures as indicated in the figure. Find x and y. Angles x and z will be complementary to 43 and y will be equal.

  2. 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. 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. Bisector of exterior angle is parallel to third side.

  4. 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 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. 1 2

  5. 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 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. 1 2

  6. 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. 2 1 Both parts follow directly from theorem1 and theorem 2. 1 2

  7. 5. Prove that parallel lines are everywhere equidistant. A B l 1 3 1 1 4 2 m C D 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.

  8. 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”.). • Construct ABC using the segment tool. • Construct ray AC and locate D on that ray so that A – C - D. • 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. • Drag point A, keeping it on (or parallel) to a fixed line through B. What happens to FCE? Does it change position and measure? • What seems to be true of ABC when ray CF is parallel to AB? • Could you prove your observations?

  9. 7. Given ABC with D on side AB and AD = DB = CD. Prove ACB = 90. C 2 1 1 2 A B D • 1 + 1 + 2 + 2 = 180 so 1 + 2 = 90

  10. 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.

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