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Transcript
|&qr!_{ix Conjectures Chapter 2 C-l Linear Pair Conjecture If two angles form a linear pair, then the nleasures of the add up C-2 angies to 180'. (Lesson 2.5) Vertical Angles Conjecture If two angles are vertical angles, then they are congruenl (have equal measures). (Lesson 2.5) C-3a Corresponding Angles Conjecture, or CA Conjecture lf two parallel lines are cut by a transversal, then corresponding angles are congruent. (Lesson 2.6) C-3b Alternate lnterior Angles Conjecture, or AIA Conjecture If two parallel lines are cut by a transversal, then alternate interior angles are congruent. (Lesson 2.6) C-3c Alternate Exterior Angles Conjecture, or AEA Conjecture If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. (Lesson 2.6) C-3 Parallel Lines Conjecture if two parallel lines are cut by a transversal, then corresponding angles are congruent, alternate interior angles are congruent, and alternate exterior angles are congruent. (Lesson 2.6) C-4 Converse of the Parallel Lines Conjecture If two lines are cut by a transversal to form pairs of congruent corresponding angles, congruent alternate interior angles, or congruent alternate exterior angles, then the lines are parallel. (Lesson 2.6) Chapter 3 C-5 Perpendicular Bisector Conjecture if a point is on the perpendicular bisector of segment, then it is equidistant from the endpoints. (Lesson 3.2) C-6 Converse of the Perpendicular Bisector Conjecture If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. a (Lesson 3.2) C-7 Shortest Distance Conjecture The shortest distance from a point to a line is measured along the perpendicular segment from the point to the line. (Lesson 3.3) C-8 Angle Bisector Conjecture If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. (Lesson 3.4) C-9 Angle Bisector Concurrency Conjecture The three angle bisectors of a triangle are concurrent (meet at a point). (Lesson 3.7) C-10 Perpendicular Bisector Concurrency Conjecture The three perpendicular bisectors of a triangle are concurrent. (Lesson 3.7) C-l1 Altitude Concurrency Conjecture The three altitudes (or the lines containing the altitudes) of a triangle are concurrent. (Lesson 3.7) C-12 Circumcenter Conjecture The circumcenter of a triangle is equidistant from the vertices. (Lesson 3.7) C-l3 lncenter Conjecture The incenter of a triangle is equidistant from the sides. (Lesson 3.7) C-14 Median Concurrency Conjecture The three medians of a triangle are concurrent. (Lesson 3.8) C-l5 122 Centroid Conjecture The centroid of a triangle divides eacl-r median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the opposite side. (Lesson 3.8) coNJECruREs Discovering Geometry Teoching and Worksheet Masters O2003 Key Curriculum Press w? rc3ml C-l6 Centerof GravityConjecture Thecentroidof atriangleisthecenterof gravityof the triangular region. (Lesson 3.8) Chapter 4 C-17 Triangle Sum Conjecture The sum of the measures of the angles in every triangle is 180'. (Lesson 4,1) C-18 Third Angle Conjecture If two angles of one triangle are equal in measure to two angles of another triangle, then the third angle in each triangle is equal in measure to the third angle in the other triangle, (Lesson 4.1) C-l9 lsosceles Triangle Conjecture If a triangle is isosceles, then its base angles are congruent. (Lesson 4.2) C-20 Converse of the lsosceles Triangle Conjecture then it is an isosceles triangle. (Lesson 4.2) If a triangle has two congruent angles, C-21 Triangle lnequality Conjecture The sum of the lengths of any two sides of a triangle is greater than the length of the third side. (Lesson 4.3) C-22 Side-Angle lnequality Conjecture In a triangle, if one side is ionger than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. (Lesson 4.3) C-23 Triangle Exterior Angle Conjecture The measure of an exterior angle of a triangle equal to the sum of the measures of the remote interior angles. (Lesson 4.3) C-24 SSS is Congruence Conjecture If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. (Lesson 4.4) C-25 5AS Congruence Conjecture If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. (Lesson 4.4) C-26 ASA Congruence Conjecture lf two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congrLrent. (Lesson 4.5) C-27 SAA Congruence Conjecture If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent. (Lesson 4.5) C-28 Vertex Angle Bisector Conjecture In an isosceles triangle, the bisector of the vertex angle is also the altitude and the median to the base. (Lesson 4.8) C-29 Equilateral/Equiangular Triangle Conjecture Every equilateral triangle is equiangular. Conversely, every equiangular triangle is equilateral. (Lesson 4.8) Chapter 5 C-30 Quadrilateral Sum Conjecture The sum of the measures of the four angles of any quadrilateral is 360". (Lesson 5.1) C-31 Pentagon Sum Conjecture The sunr of the measures of the five angies of any pentagon is 540". (Lesson 5.1) C-32 Polygon Sum Conjecture The sum of the measures of the is 180'(r - 2). (Lesson 5.1) Discovering Geometry Teaching and Worksheet Masters 02003 Key Curriculum Press r interior angles of an /,-gon coNJECTURES 123 IE@IIfiil:Gl C-33 Exterior Angle Sum Conjecture For any polygon, the sum of the measures of a set of exterior angles is 360". (Lesson 5.2) C'34 Equiangular Polygon Conjecture You can find the measure of each interior angle of an r80"(L * 2). equiangular n-gonby using either of these formulas: l80o * {I ur (Lesson 5.2) C-35 C-36 C'37 Kite Angles Conjecture The nonvertex angles of a kite are congruent. (Lesson 5.3) Kite Diagonals Conjecture The diagonals of a kite are perpendicular. (Lesson 5.3) Kite Diagonal Bisector Conjecture The diagonal connecting the vertex angles of a kite is the perpendicuiar bisector of the other diagonal. (Lesson 5.3) C-38 Kite Angle Bisector Conjecture The vertex angles of a kite are bisected by a diagonal. (Lesson 5.3) C-39 Trapezoid Consecutive Angles Conjecture The consecutive angles between the of a trapezoid are supplementary. (Lesson 5.3) C-40 lsosceles Trapezoid Conjecture The base angles of an isosceles trapezoid are congruent. bases (Lesson 5.3) C-41 lsosceles Trapezoid Diagonals Conjecture The diagonals of an isosceles trapezoid are congruent. (Lesson 5.3) C-42 Three Midsegments Conjecture The three midsegments of a triangle divide it into four congruent triangles. (Lesson 5.4) C-43 Triangle Midsegment Conjecture A midsegment of a triangle is paraliel to the third side and half the length of the third side. (Lesson 5.4) C'44 Trapezoid Midsegment Conjecture The midsegment of a trapezoid is parallel to the bases and is equal in length to the average of the lengths of the bases. (Lesson 5.4) C-45 Parallelogram Opposite Angles Conjecture The opposite angles of a parallelogram are congruent. (Lesson 5.5) C-46 Parallelogram Consecutive Angles Conjecture The consecutive angles of a parallelogram are supplementary. (Lesson 5.5) C-47 Parallelogram Opposite Sides Conjecture The opposite sides of a parallelogranl are congruent. (Lesson 5.5) C-48 Parallelogram Diagonals Conjecture The diagonals of a parallelogram bisect each other. (Lesson 5.5) C-49 Double-Edged Straightedge Conjecture If two parallel lines are intersected by a of parallel lines that are the same distance apart as the first pair, then the second pair parallelogram formed is a rhombus. (Lesson 5.6) C-50 Rhombus Diagonals Conjecture The diagonals of a rhombus are perpendicuiar, and they bisect each other. (Lesson 5,6) C-51 Rhombus Angles Conjecture The diagonals of a rhombus bisect the angles of the rhombus. (Lesson 5.6) C-52 Rectangle Diagonals Conjecture The diagonals of a rectangle are consruent and bisect each other. (Lesson 5.6) C-53 Square Diagonals Conjecture The diagonals of a sqltare are congruent, perpendicular, and bisect each other. (Lesson 5.6) 124 coNJECTUREs Discovering Geometry Teaching and Worksheet Maste$ 02003 Key Curriculum Press lWWit Chapter 5 c-54 ' chord central Angles conjecture If two chords in a circle are congruent, then they determine two central angles that are congruent. (Lesson 6.1) c-55 chord Arcs conjecture If two chords in a circle are congruent, then their intercepted are congruent. (Lesson 6.1) c-56 c'57 c-58 arcs Perpendicular to a chord conjecture The perpendicular from the center of a circre to chord is the bisector of the chord. (Lesson 6.li a chord Distance to center conjecture Two congruent chords in a circle are equi<listant from the center of the circle. (Lesson 6.1) Perpendicular Bisector of a chord coniecture The perpendicular bisector of a chord through the center of the circle. (Lesson 6.1) Tangent conjecture A tangent to a circle is perpendicular to the radius drawn to the point of tangency. (Lesson 6.2) passes c-59 c-60 Tangent segments conjecture Thngent segments to a circle from a point outside the circle are congruent. (Lesson 6.2) c'61 c-62 Inscribed Angle conjecture The measure of an angle inscribed in a circle is one-half the measure of the central angle. (Lesson 6.3) lnscribed Angtes lntercepting Arcs conjecture Inscribed angles that intercept the same arc are congruent. (Lesson 6.3) c-53 Angles lnscribed in a semicircle conjecture Angles inscribecl in a semicircie are right angles. (Lesson c-64 cyclic Quadrilateral conjecture The opposite angres of a cyclic quadrilateral 6.3) supplementary. (Lesson 6.3) c-65 Parallel Lines lntercepted Arcs conjecture Paraliel lines a circle. (Lesson 6.3) c-66 are intercept congruent arcs on circumference conjecture If c is the circumference and d is the diameter of a circle, then thereisanumberz-suchthat c: rd. If cr:2rwhere ristheradius,then c :2nr. (Lesson 6.5) c-67 Arc Length conjecture The tength of an arc equals the circumference times the measure of the cenrral angle divided by 36-0". (Lesson 6.7f Chapter 7 c-58 Reflection Line coniecture The line of reflection is the perpendicular bisector of every segment joining a point in the originar figure with its i.,rug.. (Lesson 7.r) C-69 Coordinate Transformations Conjecture The ordered pair rule (x, y) --> (-x, y) is a reflection over the 7_axis. The ordered pair rule (x, y) _-s (x, _y) is a reflection over the x-axis. The ordered pair rure (x, y)-+(-x, *y) is a rotation about trre origin. The ordered pair rule (x, y) _-> (Z x) is a reflection Discovering Geometry Teaching and Worksheet Masters O2003 Key Curriculum press over y: x. (Lesson 7.2) coNJECTURES lZS