FTCE Middle Grades Math 5-9 Angles Competency 8 Angle pairs formed when two parallel lines are cut by a transversal Complementary angles – add up to 90° 1 3 5 7 2 4 6 8 Corresponding Angles - angles at the same location at each intersection. (Congruent) Supplementary angles – add up to 180° ∠1 and ∠5 ∠3 and ∠7 ∠2 and ∠6 ∠4 and ∠8 Alternate Interior Angles - two interior angles which lie on different parallel lines and on opposite sides of a transversal. (Congruent) ∠4 and ∠5 Adjacent angles – have a common vertex and share a side ∠3 and ∠6 Alternate Exterior Angles - two exterior angles which lie on different parallel lines and on opposite sides of a transversal. (Congruent) ∠1 and ∠8 ∠2 and ∠7 Lines Parallel lines – two coplanar lines that do not intersect (they have the same slope. Vertical angles – angles opposite each other when two lines cross. (Congruent) Perpendicular lines – two lines that intersect to form right angles (slopes are negative reciprocals of each other. Skew lines – two noncoplanar lines that do not intersect. FTCE Middle Grades Math 5-9 Competency 8 Classifying Triangles Properties of Congruence Sides Reflexive Property A quantity is congruent (equal) to itself. a ≅ a Equilateral – three congruent sides Symmetric Property If a ≅ b, then b ≅ a. Transitive Property If a ≅ b and b ≅ c, then a ≅c. Isosceles – at least two congruent sides Sum of the measures of the interior angles of a polygon: 180°(n-2) Base angles are ≅ Scalene – no congruent sides n = the number of sides of the polygon Measure of each interior angles of a regular polygon: 180°(n-2) n Angles Sum of the measures of the exterior angles of a polygon = 360 Acute – three acute angles Triangle Inequality Theorem The sum of the lengths any two sides of a triangle must be greater than the length of the third side. Right - one right angle a+b>c b+c>a a+c>b Triangle Inequalities – in a triangle the largest angle is opposite the longest side and the smallest angle is opposite the shortest side. Obtuse – one obtuse angle FTCE Middle Grades Math 5-9 Pythagorean Theorem Competency 8 Special Right Triangles a2 + b 2 = c2 45° - 45° - 90° Triangle Congruency Theorems 30° - 60° - 90° Right Triangle Trigonometry “SOHCAHTOA” Similar Figures - Corresponding angles are congruent and corresponding sides are proportional. 10 12 5 = = 20 24 10 FTCE Middle Grades Math 5-9 Distance Formula Competency 8 Defined Terms Line Segment - is a part of a line that is bounded by two end points. Midpoint Formula Undefined Terms Point – a location, it has no width, length, or height Line – an infinite number of points. It is determined by two points. Name a segment with it’s endpoints and a segment above them. Ray - the part of the line which consists of the given point and the set of all points on one side of the end point. Name a ray with it’s endpoint and any other point on the ray with a one sided arrow above it. Plane – flat 2D surface extending infititely in two directions. It is defined by three noncollinear points. Collinear points – points that lie on the same line Non-collinear points – points that do not lie on the same line Name a line using any two points on the line with a double arrow above it. Coplanar points – points that lie on the same plane Symmetry Non-coplanar points – points that do not lie on the same plane (A, B, C, and P are non-coplanar) A line of symmetry divides a figure into to halves that are the mirror images of each other. FTCE Middle Grades Math 5-9 Transformations Competency 8 Tessellation - A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. Rotation To rotate an object means to turn it around. Every rotation has a center and an angle. Translation To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance. Three Dimensional Figures Prism – a polyhedron with two parallel bases and rectangular faces. Reflection To reflect an object means to produce its mirror image. Every reflection has a mirror line. A reflection of an "R" is a backwards "R". Glide Reflection A glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step. Dilation A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure. Pyramid– a polyhedron with one base and triangular faces. FTCE Middle Grades Math 5-9 Competency 8 Area Formulas Area of a rectangle = base ∙ height Area of a triangle = ½ base ∙ height Area of a trapezoid = ½ (base1 + base 2) ∙ height Circles Cylinder – A cylinder is similar to a prism, but its two bases are circles, not polygons. Cone - cone has one circular base and a vertex that is not on the base. Sphere – all points on the sphere are equidistant from the center. Platonic Solids Radius - the distance from the center of the circle to the outside edge. Chord - a line segment that connects one point on the edge of the circle with another point on the circle. Diameter – a chord that goes through the center of a circle. Arc - segment of the circumference of the circle. Circumference - the distance around the circle Circumference = 2πr where r = the radius of the circle The formula for the arc length of a circle: θ 360 ⋅ 2π r Area = πr2 Sector – area of a circle bounded by two radii and an arc. Euler’s Formula Segment – area of circle bounded by a chord and an arc. For a polyhedron the number of vertices + faces = the number of edges + 2 V+F=E+2 Constructions http://www.mathsisfun.com/geometry/constructio ns.html Line Bisector (perpendicular bisector) Angle Bisector Perpendicular to a Point on a Line Same (Congruent) Angle The formula for area of a sector of a circle: θ 360 ⋅π r 2 FTCE Middle Grades Math 5-9 Angles in Circles Central Angle - an angle formed by two intersecting radii such that its vertex is at the center of the circle. Central angle = measure of intercepted arc ∠x = 80° Inscribed Angle - an angle with its vertex "on" the circle, formed by two intersecting chords. Competency 8 Angle Formed Inside of a Circle by Two Intersecting Chords: When two chords intersect "inside" a circle, four angles are formed. At the point of intersection, two sets of vertical angles can be seen in the corners of the X that is formed on the picture. angle = ½ sum of the intercepted arcs ∠x = ½ (170+70)°= 120° Angle Formed Outside of a Circle by the Intersection of: "Two Tangents" or "Two Secants" or "a Tangent and a Secant" angle = ½ difference of the intercepted arcs Two Tangents Inscribed angle = ½ measure of intercepted arc ∠x = 50° Tangent Chord Angle - an angle formed by an intersecting tangent and chord has its vertex "on" the circle. ∠x = ½ (260-100)°= 80° Two Tangents ∠x = ½ (80-20)°= 30° Tangent Chord angle = ½ measure of intercepted arc ∠x = 60° A Tangent and a Secant ∠x = ½ (100-30)°= 35° FTCE Middle Grades Math 5-9 Competency 8 Segments in Circles Intersecting chords -‐ the product of the lengths of the segments of one chord equal the product of the segments of the other. Two secants drawn from the same point -‐ the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part. Secant and tangent from the same point -‐ the product of the length of the secant segment and its external part equals the square of the length of the tangent segment.