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(2 x +10) °

G. •. B. In the diagram, AB CD. The measures of GED and GFB are shown. Find the value of the variable x . Find m(BFH). E. In the diagram, AB CD, and PQ bisects CPB. If m(PCQ) = 38 °, what is the measure of PQD?. •. •. •. P. D. C. D. A. •. •. •. Q. A. F. B. C.

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(2 x +10) °

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  1. G • B In the diagram, AB CD. The measures of GED and GFB are shown. • Find the value of the variable x. • Find m(BFH) E In the diagram, AB CD, and PQ bisects CPB. If m(PCQ) = 38°, what is the measure of PQD? • • • P D C D A • • • Q A F B C • H (2x+10)° (3x – 13)° x = 23 124 • • • • • 108 •

  2. From HW # 2 7. Using Geometer’s Sketchpad a. Construct the diagram shown below, in which two pairs of line segments are parallel. (The letters a, b, and x do not need to appear on the sketch.) Please include a hard-copy of the sketch when you turn in this assignment. b. Use the measurement tool to display the measures of the angles marked a, b and x. c. Make a conjecture about the relationship between the measures of the three angles. d. Explain why your conjecture will always be true. x b a

  3. I'm working on problem 7 on HW2 and this is what it looks like so far: I was wondering how to cut the parts of the segment that are below the horizontal and the parts above the endpoints. 

  4. From HW # 2 x = 131 x = 48 x = 20 x = 22.5

  5. From HW # 2 Paper and pencil constructions 5. Construct a parallel through point S to line m.

  6. From HW # 2 Paper and pencil constructions 6. Construct a triangle two of whose three angles have the same measure as A.

  7. From HW # 2 Conjecture: a + b – x = 180 7. Using Geometer’s Sketchpad a. Construct the diagram shown below, in which two pairs of line segments are parallel. (The letters a, b, and x do not need to appear on the sketch.) Please include a hard-copy of the sketch when you turn in this assignment. b. Use the measurement tool to display the measures of the angles marked a, b and x. c. Make a conjecture about the relationship between the measures of the three angles. d. Explain why your conjecture will always be true. E F D C x b a B A P

  8. B A C D From HW # 2 8. Using Geometer’s Sketchpad a. Construct the diagram below, in which is parallel to . Your diagram should look exactly like the one shown. b. Use the measurement tool to display the measures of BCD, ADC, A, and B. c. Display the sum of the measures of these four angles. d. Use your cursor to move point B. Does the sum change?

  9. Conjecture: The measures have a sum of 180 B A C D From HW # 2 8. Using Geometer’s Sketchpad a. Construct the diagram below, in which is parallel to . Your diagram should look exactly like the one shown. b. Use the measurement tool to display the measures of BCD, ADC, A, and B. c. Display the sum of the measures of these four angles. d. Use your cursor to move point B. Does the sum change? 4 3 5 6 1 2

  10. 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?

  11. If two parallel lines are cut by a transversal, then corresponding angles are congruent. Converse If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

  12. Converses of Parallel Lines Theorems If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. If two lines are cut by a transversal and same side interior angles are supplementary, then the lines are parallel. If a transversal is perpendicular to each of two lines, then the two lines are parallel.

  13. More Basic Constructions

  14. P R Q B A Basic Construction 3: Steps for constructing a parallel to a line l(or AB)through a point P not on the line. 1. Construct a line through P that intersects line AB at point Q. 2. Follow the steps for “copying an angle” to construct an angle QPR that is congruent to PQB and having as one of its sides. Conclusion: is parallel to l

  15. A P B N Q C Basic Construction 4: Stepsfor 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.

  16. Using Geometer’s Sketchpad, construct the diagram shown, in which AB is parallel to CD, MP bisects BMN, and NP bisects DNM.

  17. Using Geometer’s Sketchpad, construct the diagram shown, in which AB is parallel to CD, MP bisects BMN, and NP bisects DNM. Make a conjecture about the measure of P. Explain why your conjecture will always be true.

  18. How can we be sure that our conclusion is correct? (in other words, why does this process guarantee that SPR is congruent to ABC? Basic Construction 2: Copying a given angle ABC: • “Construct” a ray . • Construct a circle of convenient radius with point B as center. Call the intersection of the circle with point M and the intersection of the circle with point N. • Construct a congruent circle with point P as center. Call its intersection with point R. • Construct a circle with center M and radius . • Construct a circle congruent to the one in step 5 with R as center. Call the intersection of this circle and circle P, point S. • “Construct” . Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.

  19. Congruent triangles are triangles with all pairs of corresponding sides and all pairs of corresponding angles congruent. ABC  EDF

  20. The three triangle congruence postulates: If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent (SSS). If two sides of one triangle are congruent to two sides of another triangle, and the angles between these pairs of sides are congruent, then the triangles are congruent (SAS). If two angles of one triangle are congruent to two angles of another triangle, and the sides between these pairs of angles are congruent, then the triangles are congruent (ASA). Corresponding parts of congruent triangles are congruent. (CPCTC)

  21. Proving that the basic constructions do what we claim they do.

  22. Basic Construction 2: Copying a given angle ABC: • “Construct” a ray . • Construct a circle of convenient radius with point B as center. Call the intersection of the circle with point M and the intersection of the circle with point N. • Construct a congruent circle with point P as center. Call its intersection with point R. • Construct a circle with center M and radius . • Construct a circle congruent to the one in step 5 with R as center. Call the intersection of this circle and circle P, point S. • “Construct” . Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.

  23. Proof of the construction BN  PR, and BM  PS because they are radii of congruent circles. Similarly, MN  SR. Therefore, MBN  SPR (SSS) and SPR is congruent to ABC (CPCTC).

  24. 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.

  25. 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).

  26. P B A Q

  27. 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

  28. 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

  29. Homework: Download, print, and complete Homework # 3 (Constructions Continued)

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