Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Coxeter notation wikipedia , lookup
Multilateration wikipedia , lookup
Shapley–Folkman lemma wikipedia , lookup
Perceived visual angle wikipedia , lookup
Poincaré conjecture wikipedia , lookup
Four color theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Integer triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Pythagorean theorem wikipedia , lookup
From HW # 5 1. In class we discussed that the sum of the measures of the interior angles of a convex quadrilateral is 360o.” Using Geometer’s Sketchpad, determine whether this is true for a concave quadrilateral. Please display all appropriate angle measures and sums. 2. Another theorem that we stated in class said ”The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360o.” Using Geometer’s Sketchpad, determine whether this is true for a concave quadrilateral. Please display all appropriate angle measures and sums. 3. In the diagram at the right, all the vertices of quadrilateral ABCD lie on a circle (we say that quadrilateral ABCD is , inscribed in the circle.). a. Construct the diagram using Geometer’s Sketchpad. b. Make a conjecture about how the measures of angle and angle BCD are related. c. Drag one or more of the quadrilateral’s vertices and verify your conjecture (or form a new conjecture if your first conjecture turns out to be incorrect). A D B C From HW # 5 4. In the diagram, segments AD, BE , and CF are concurrent (intersect at point P). What is the sum of the measures of the six “corner” angles? (i.e. the sum of the measures of angles A, B, C, D, E, and F) B A 360° C P F D E From HW # 5 5. In the diagram, AE is parallel to BF, and DE is parallel to FC. Prove that E F. E F A B C D From HW # 5 6. In the diagram, four right triangles are shown, and PQ and RQ are perpendicular. What is the sum of the measures of the four numbered angles? 270° P 1 2 3 4 Q R From HW # 5 7. In the diagram, points A, B, and C are collinear. Express the sum of the measures of the five numbered angles as a function of x. x + 360° 4 3 2 1 A x 5 B C From HW # 5 8. In the diagram, what is the sum of the measures of the six “corner” angles (i.e. the sum of the measures of angles A, B, C, D, E, and F)? B C 360° A E F D From HW # 5 8. In the diagram, what is the sum of the measures of the six “corner” angles (i.e. the sum of the measures of angles A, B, C, D, E, and F)? B C 360° A E F D From HW # 5 8. In the diagram, what is the sum of the measures of the six “corner” angles (i.e. the sum of the measures of angles A, B, C, D, E, and F)? B C 360° A E F D Practice Problems B 1. Find the sum of the measures of A, B, C, D, and E A 40 C E D B 2. In the diagram, BE bisects ABC. What is the measure of BED? A 48 108 96 D E C Practice Problems B 1. Find the sum of the measures of A, B, C, D, and E A 40 460 C E D B 2. In the diagram, BE bisects ABC. What is the measure of BED? A 48 108 102 96 D E C 3. In the diagram, compute the sum of the angles numbered 1 through 8. 1 7 5 2 6 8 4 3 3. In the diagram, compute the sum of the angles numbered 1 through 8. 1 720 7 5 2 6 8 4 3 The three congruence postulates we have are SSS, SAS, and ASA. An additional method for proving triangles congruent AAS The Isosceles Triangle Theorem (Base Angles Theorem) If a triangle has two congruent sides, the angles opposite those sides are congruent. The converse is also true: If a triangle has two congruent angles, the sides opposite those angles are congruent. leg leg base Application 1 1. In the diagram, ABC is isosceles with AB = BC, and AP is an altitude. If the measure of angle B is 48, what is the measure PAC? B 48 P A 24 C B P E A D C Another triangle congruence theorem: hypotenuse - leg (HL) Can two triangles be proven congruent by SSA ASS ? A counter-example to SSA These triangles agree in SSA, yet they cannot be congruent. X • Moral: If you use ASS to prove triangles congruent, you… 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 AB . Label its intersection with the angle bisector point D. d. Make a conjecture about the relationship between the length of AC and the length of CD. ItProve is notyour necessary to prove your conjecture. conjecture. Conjecture: AC CD B D A C (n 3)(n) A convex polygon with n-sides has diagonals. 2 (n 3)(n) A convex polygon with n-sides has . 2 Homework: Download, print, and complete Homework # 6 Theorem: Every right angle is obtuse. Here is the “proof” of the theorem: A D E B Given: Right angle ABC and obtuse angle DCB Prove: ABC DCB F C 1. Using a compass, construct points E and F on ray BA and ray CD so that BE CF. A D E C B Question: Suppose they were parallel. F A D E C B Question: Suppose they were parallel. F A D E B 3. C F A E B 4. D C F A D E B 4. C F A D E B M N F C 5. The perpendicular bisectors are not parallel. 4. P labeling the midpoints N and M, respectively. 6. Call their point of intersection, P A D E B M P 7. Construct segments EP, FP, BP, and CP N F C 8. PEN PFN SAS 9. BPM CPM SAS 10. EP FP and BP CP A D E B M P N F C 12. EBP FCP SSS A D E B M N F C Therefore, right angle ABC is congruent to obtuse angle FCB. Subtract the measures of the green angles from the measures of the red angles. P Where is the flaw in the proof?