FTCE Middle Grades Math 5-9 Competency 8

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.