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Warm-up Using Geometer’s Sketchpad, construct a rectangle whose length

If you do not have YES next to your student #, either you did not send me GSP constructions or I am waiting for corrections. Warm-up Using Geometer’s Sketchpad, construct a rectangle whose length and width are in the ratio of 2:1 and display the ratio.

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Warm-up Using Geometer’s Sketchpad, construct a rectangle whose length

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  1. If you do not have YES next to your student #, either you did not send me GSP constructions or I am waiting for corrections.

  2. Warm-up • Using Geometer’s Sketchpad, construct a rectangle whose length • and width are in the ratio of 2:1 and display the ratio. Using compass and straight edge, construct a right angle using only the angle bisector construction (Basic Construction 4).

  3. Two angles form a linear pairif they are adjacent and their exterior sides are opposite rays. Theorem: The angles of a linear pair are supplementary. Theorem: If two angles are congruent and supplementary, they are right angles.

  4. From HW # 3 1. Using Geometer’s Sketchpad a. Construct triangle ABC. b. Construct the angle bisector of BAC c. Construct a line through point C parallel to . Label its intersection with the angle bisector point D. d. Make a conjecture about the relationship between the length of and the length of . It is not necessary to prove your conjecture. Conjecture:

  5. B Conjecture: the length of is three times the length of . P M C A D

  6. From HW # 3 4. Construct a triangle congruent to triangle ABC.   C A B

  7. C  D  1 2 3  B  A • Use Geometer’s Sketchpad to construct the following diagram, in which line DC is parallel to line AB and point Q is randomly chosen between them.  Q 2. Display the measures of <1, <2, and <3 3. Make a conjecture about how the three measures are related to one another. Conjecture: m2 = m1 + m3 4. Drag point Q and verify your conjecture or form a new conjecture. 5. Can you prove the conjecture?

  8. F  E Conjecture: m2 = m1 + m3 C  D 1 + 3 2 = red + blue  1  Q 2 3  B  A

  9. How can we be sure that our conclusion is correct? Basic Construction 4: Constructing the bisector of a given angle ABC. 1. Construct a circle using point B as center, intersecting at point P and at point Q. 2. Construct congruent circles with centers at P and Q. Use a radius that will cause the two circles to intersect. Call the intersection point N. 3. Construct . Conclusion: is the bisector of ABC.

  10. Proof of the construction BP  BQ because they are radii of congruent circles. Similarly, PN  QN. Since BN  BN (Reflexive Postulate), PBN  QBN (SSS) and PBN is congruent to QBN (CPCTC).

  11. P B A Q

  12. Q A B P Basic Construction 6: Steps for constructing a perpendicular to a line l through a point P on the line. 1. Construct a circle with center at point P intersecting line l in two points, A and B. 2. Construct congruent circles with centers at A and B, and radii at least as long as . 3. Call the intersection of the two congruent circles, point Q. 4. Construct . Conclusion: is perpendicular to line l. l

  13. Basic Construction 7: Steps for constructing a perpendicular to a line l through a point P not on the line. 1. Construct a circle with center at point P intersecting line l in two points, A and B. 2. Construct congruent circles with centers at A and B, and radii at least as long as . 3. Call the intersection of the two congruent circles, point Q. 4. Construct . Conclusion: is perpendicular to line l. P B A l Q

  14. P B A l Q

  15. (Prove: AMP and BMP are right angles) They are radii of congruent circles Radii of congruent circles are congruent. Reflexive property SSS 4. PAQ  PBQ P 5. APQ  BPQ CPCTC Same as 3 7. PAM  PBM SAS M B A l 8. AMP  BMP CPCTC 9. AMP is supplementary to BMP. The angles of a linear pair are supplementary If two angles are congruent and supplementary, they are right angles. 10. AMP and BMP are right angles. Q

  16. Theorems that should make perfect sense to you 1. If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent. 2. If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent. 3. Vertical angles are congruent. 4. If two lines intersect, then they intersect in exactly one point. 5. Every segment has exactly one midpoint. 6. Every angle has exactly one bisector.

  17. Proof: Construct parallel to . Q P PCA QCB ACB + + The sum of the measures of the angles of a triangle is 180°. C B A B A = 180°

  18. Related Corollaries and Theorems • Through a point outside a line, exactly one perpendicular can be drawn …to the line. P • l

  19. Related Corollaries and Theorems • Through a point outside a line, exactly one perpendicular can be drawn …to the line. • If two angles of one triangle are congruent to two angles of another …triangle, then the third angles are congruent. • Each angle of an equiangular triangle has measure 60o. • In a triangle, there can be at most one right angle or one obtuse angle. • The acute angles of a right triangle are complementary.

  20. D A B C In the diagram, ABD  DCA and . If the measure of DCB is 50, what is the measure of A?

  21. Related Corollaries and Theorems • Through a point outside a line, exactly one perpendicular can be drawn …to the line. • If two angles of one triangle are congruent to two angles of another …triangle, then the third angles are congruent. • Each angle of an equiangular triangle has measure 60o. • In a triangle, there can be at most one right angle or one obtuse angle. • The acute angles of a right triangle are complementary. • If one side of a triangle is extended, then the measure of the exterior …angle(s) formed is equal to the sum of the measures of the two remote …interior (non-adjacent interior) angles.

  22. If one side of a triangle is extended, then the measure of the exterior angle(s) formed is equal to the sum of the measures of the two remote interior (non-adjacent interior) angles.

  23. C  D  1 2 3  B  A Last class, we used Geometer’s Sketchpad to investigate the following problem. • Use Geometer’s Sketchpad to construct the following diagram, in which line DC is parallel to line AB and point Q is randomly chosen between them.  Q 2. Display the measures of <1, <2, and <3 3. Make a conjecture about how the three measures are related to one another. Conjecture: m2 = m1 + m3 4. Drag point Q and verify your conjecture or form a new conjecture. 5. Can you prove the conjecture?

  24. F  E Conjecture: m2 = m1 + m3 C  D 1 + 3 2 = red + blue  1  Q 2 3  B  A

  25. Homework: Download, print, and complete Homework # 4

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