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Conjecture- unproven statement that is based on observations. Inductive reasoning- looking for patterns and making conjectures is part of this process Counterexample- example shows a conjecture is false. Point- no dimensions. Line- extends in one dimension. Plane- extends in two directions. Collinear points- points that lie on the same line. Coplanar points- points that lie on the same plane. Acute angles- less than 90 degrees Right angles- equal to 90 degrees Obtuse angles- more than 90 degrees Straight angles- equal to 180 degrees Complementary angles- sum of measures is 90 degrees. Supplementary angles- sum of measures is 180 degrees. Distance Formula- › AB=Square root of(x-x1)²+(y2-y1)² Pythagorean Theorem- a²+b²=c² Parallel lines-coplanar and do not intersect. Transversal- a line that intersects two or more coplanar lines at different points. Corresponding angles- if they occupy corresponding positions. Alternate exterior angles- if they lie outside the two lines on opposite sides of the transversal. Alternate interior angles- if they lie between the two lines on opposite sides of the transversal. Consecutive interior angles- if they lie between the two lines on the same side of the transversal. Theorem 3.1 › If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Theorem 3.2 › If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Theorem 3.3 › If two lines are perpendicular, then they intersect to form four right angles. Theorem 3.4- Alternate Interior Angles › If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Theorem 3.5- Consecutive Interior Angles › If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Theorem 3.6- Alternate Exterior Angles › If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Vertex- each of the three points joining the sides of a triangle. Adjacent sides- two sides sharing a common vertex. Hypotenuse- side opposite of the right angle. Congruent- correspondence between their angles and sides. Names of Triangles › Equilateral › Isosceles › Scalene Classification by angles › Acute › Equiangular › Right › Obtuse Reflexive Property › Every triangle is congruent to itself. Symmetric Property › If triangle ABC is congruent to triangle DEF, the triangle DEF is congruent to triangle ABC. Transitive Property › If triangle ABC is congruent to triangle DEF and triangle DEF is congruent to triangle JKL, then triangle ABC is congruent to triangle JKL. Side- Side-Side Side-Angle-Side Angle-Side-Angle Angle-Angle-Side Convex- if no line that contains a side of the polygon contains a point in the interior of the polygon. Concave- a polygon that is not convex. Diagonal- a segment that joins two nonconsecutive vertices. Theorem 6.6 › If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6.7 › If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6.8 › If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Preimage- original figure. Image- new figure Transformation- operation that moves the preimage to the image. Translation- a transformation that maps every two points. Theorem 7.2 Rotation Theorem › A rotation is an isometry. Theorem 7.3 › If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is a rotation about point P. Proportion- an equation that equates two ratios. Geometric mean- two positive numbers a and b is the positive number x such that a/x=x/b. Similar polygons- when there is a correspondence between two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional. Side-Side-Side › If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. Side-Angle-Side › If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. Special right triangles- have measures of 45-45-90 or 30-60-90. Sin, cosine, tangent- three basic trigonometric ratios. Trigonometric ratio- ratio of the lengths of two sides of a right triangle. Theorem 9.8 45-45-90 Triangle › In a 45-45-90 triangle, the hypotenuse is square root of 2 times as long as each leg. Theorem 9.9 30-60-90 Triangle › In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is square root of 3 times as long as the shorter leg. Diameter- distance across the circle. Radius- distance from center to point on the circle. Chord- segment whose endpoints are points on the circle. Secant- a line that intersects a circle in two points. Tangent- a line in the plane of a circle that intersects the circle in exactly one point. Minor arc- part of a circle that measures less than 180 degrees. Major arc- part of a circle that measures between 180 degrees and 360 degrees. Semicircle- if the endpoints of an arc are the endpoints of diameter, then it is a semicircle. Theorem 10.1 › If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Theorem 10.2 › In a plane, if a line is perpendicular to a radius if a circle at its endpoint on the circle, then the line is tangent to the circle. Circumference- is the distance around the circle. Arc length- portion of the circumference of a circle. Semicircle- one half of the circumference. To find the sum of the measures of interior angles of a polygon › 180 multiplied by the number of sides. Given that the radius of the circle is 5 cm, calculate the area of the shaded sector. (Take π = 3.142). Area of Sector= =13.09cm² Finding Arc Length Or you can use this step: Theorem 11.4 – Area of Regular Polygons › The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P, so A=(1/2)aP or A=(1/2)aXns. Theorem 11.8 › The ratio of the area A of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360 degrees. Faces- a solid that is bounded by polygons. Edge- line segment formed by the intersection of two faces. Vertex- point where three or more edges meet.