Geometry
How to Find the Degrees of One Interior Angle
(n-2)(180)/n
Going Backwards
(n-2)(180)=________ Solve using algebraaaaa!!!
One Secant and One Tangent Case
(whole segment) (outer part) = (tangent)^2
Two Secant Case
(whole segment) (outer part) = (whole segment) (outer part)
Theorems and Postulates about Logic
Law of Syllogism: If both conditionals are true, then you may conclude correctly, that given Hypothesis 1, Conclusion 2 is true.
Skew lines
Lines that cannot be captured by a single plane
Orientation
Make sure to orient the figures in the same way because, when figures are oriented properly, it is more likely to compare parts and angles that are corresponding.
Translation
Maps all points in a figure the same distance in the same direction (isometries) *A composition of transformations means more than one
Reflections
Maps geometric objects across a line of reflection (isometries) *if r is the x-axis, it would be (x,y)->(x,-y) *if r is the y-axis, it would be (x,y)->(-x,y)
Rotations
Maps geometric objects into a rotation (isometries) * has a center, an angle, and a direction *90°=quarter, 180°=half, 360°=full
Dialations
Maps geometric objects into dilated forms *|SF|>1: Enlargement *|SF|=1: Isometry *|SF|<1: Reduction
Minor Arcs, Major Arcs
Minor Arc<180° Major Arc>180°
Arcs
An unbroken collection of points on a circle
Theorems about Circles con3.
13)If two chords intersect in a circle, the angles formed equal the average of the two intercepted arcs. 14) The angle formed by two secants, a secant and a tangent, or two tangents equals one half the difference of the larger intercepted arc minus the smaller. 15) If two chords intersect in a circle, the products of each chord's segments are equal.
Diagonals of a Polygon
A segment that goes from one vertex to a non-adjacent vertex
Geometric Mean
A special number which lies between two positive numbers *ALWAYS appears two times in the equation 4/x=x/16 (4=First number, x=geometric mean, 16=second number) - 1 Part of Hypot./Altitude of Hypot.=Altitude of Hypot./Other Part of Hypot.
Not Prove Congruence
AAA: Size might change SSA: The angle of the other two can change even if the side length stays the same
Solving a Triangles
- Finding the measurements of all the sides and angles _______________________________________________________________________ | Given | Finding Sides | Finding Angles | -------------------------------------------------------- | Two sides| Pythagorean | Trig (backwards) | -------------------------------------------------------- | One side, one angle|Trig|Subtraction | --------------------------------------------------------
Geometric Mean with Legs
- Left Leg Case: Left Part of Hypot./Left Leg = Left Leg/(Whole) Hypotenuse - Right Leg Case: Right Part of Hypot./Right Leg = Right Leg/(Whole) Hypotenuse
Measuring Angles
- Lengths have nothing to do with the size of an angle - Size of an angle measures rotation with the vertex being the center of the angle - Measure = amount of rotation around the vertex beginning at one ray (initial ray) and ending at the other (terminating ray) - identified with ° symbol - Complete rotation = 360° - "m" = degree measure
Grouping by Size
- Measure of all angles in triangle is less than 90° = acute triangle - Measure of one angle in triangle is 90° = right triangle - Measure of one angle in triangle is more than 90° = obtuse triangle
Segments+Triangles
- Median: a line segment that goes from a vertex to the middle of the opposite side (divides opposite side into two equal parts) - Centroid: the point where three medians of a triangles meet (2/3 of the way from each vertex) - Altitude: perpendicular line segment that goes from a vertex to the line that includes the opposite ray
Parallelograms
- Named by their vertices - Both pairs of opposite sides are parallel - Both pairs of opposite sides are equal - The diagonals divide each other in half - Diagonals of parallelograms bisect each other - Add up to 360° - Any two consecutive angles of a parallelogram are supplementary!
Perpendicular Bisectors and Angle Bisectors
- Perpendicular Bisectors: a segment, ray or line which is perpendicular to a segment at its midpoint - Angle Bisectors: A ray that divides an angle into two congruent adjacent angles
if then
- Premise: if (p) - Conclusion: then (q) - If there is one example that counters the premise, then it is false (this is called the counterexample)
Special Parallelograms
- Rectangle: A quadrilateral with four right angles - Rhombus: A quadrilateral with four congruent sides - Square: A quadrilateral with four congruent sides and four right angles (All squares are rectangles but not all rectangles are squares) (All squares are rhombi but not all rhombi are squares)
Adjacent
- Right next to - If any three adjacent parts of one triangle are congruent to three adjacent parts of another triangle, the two triangles are congruent
Similarity
- Scale models are smaller but otherwise exact copies of the full-sized project which they represent - Similarity has corresponding angles that are congruent and corresponding sides that are in proportion - Symbol = ~
Categories
- Scalene: The sides of a triangle have all different lengths - Isosceles: At least two of the sides of a triangle have the same length - Equilateral: The lengths of all three sides are equal
Tangents
- Straight lines - Perpendicular to a radius or diameter drawn to the point of tangency
Segment Lengths and Parallel Lines
- The distance between two parallel lines is the perpendicular distance - If three or more parallel lines cut off equal segments on a transversal, then they are the same distance apart.
Median of a Trapezoid
- The segment that connects the midpoints of the legs - The length of it is equal to the average length of its bases (median=base1+base2/2)
Sum of Triangles
- The two smaller lengths HAVE to be bigger than the longer length to be a triangle - To find the length of an unknown side, use this equation: BigNumber-SmallNumber < Length of Third Side < BigNumber+SmallNumber
Special Pairs of Angles
- Vertical Angles: Two opposite angles formed by the intersection of two lines - Adjacent Angles: A pair of angles that have the same vertex, share a common side, and do not overlap - Linear pair: Two angles where the non-shared sides are opposite rays and the angles total 180°
Angles and the Arcs They Intercept
- When the vertex of an angle is placed at the center of a circle, the sides of the angle intercept the circle and determine an arc - The measure of a minor arc equals the measure of its central angle - If more than one inscribed angle intercept the same arc, the angles must be equal - Central angle = measure of arc - Inscribed angle = 1/2 measure of arc - Chord/tangent = 1/2 measure of arc - Intersection of two chords = Arc+Arc / 2
Proportions
- When two ratios are equal, an equation called a proportion is formed - extreme/mean=mean/extreme
Congruent in Triangles
- ~ is same shape - = is same size - same shape and same size
Practical Applications of Trigonometry
------ = horizon
Properties of Proportion
0) Given: a/b=c/d 1) Means Extremes Property: ad=bc 2) Flip the ratios: b/a=d/c 3) Swap the means: a/c=b/d 4) Swap the extremes: d/b=c/a 5) Bring up the denominator: a+b/b=c+d/d 6) Add them up: a+c/b+d=a/b
Special Names for Special Polygons
1) 3 sides -- triangle 2) 4 sides -- quadrilateral 3) 5 sides -- pentagon 4) 6 sides -- hexagon 5) 8 sides -- octagon 6) 10 sides -- decagon 7) 12 sides -- dodecagon
Angle Bisector
The ray that divides an angle into two congruent adjacent angles
Semiperimeter
The semiperimeter of a triangle with sides a, b, and c is: S=a+b+c ------- 2
Theorem: Special Right Triangles
1) 45°-45°-90° Triangles: Hypotenuse=x√2 (if x are the legs) note: √2 is approximately 1.4 note: a diagonal divides a square into two congruent 45°-45°-90° triangles 2) 30°-60°-90° Triangles: - side opposite of the 60° angle = x√3 - side opposite of the 90° angle = 2x - side opposite of the 30° angle = x note: √3 is approximately 1.73
Area
The size of its surface Formula: A=πr^2
More Tangents
- Externally Tangent Circles: A point of tangency with two circles separated only by a line in between - Internally Tangent Circles: A point of tangency with two circles separated only by a line on the shared edges
Law of Cosines
The square of one side of a triangle is related to the other two sides and the included angle of the triangle a^2 = b^2 + c^2 - 2bc(cosA) b^2 = a^2 + c^2 - 2ac(cosB) c^2 = a^2 + b^2 - 2ab(cosC) (Use to solve a triangle when you do not have a pair)
Rectangles
- 4 congruent right angles - Diagonals are congruent - Diagonals bisect each other - Opposite sides are congruent - Opposite sides are parallel - If it has one right angle, then it will have four right angles
Square
- 4 congruent sides - 4 congruent right angles - Diagonals are congruent - Diagonals are perpendicular - Diagonals form four right angles - Diagonals bisect the right angles into 45° angles - Diagonals bisect each other - Opposite sides are parallel
Rhombus
- 4 congruent sides - Diagonals are perpendicular - Diagonals bisect angles - Diagonals form four congruent right angles - Opposite angles are congruent - Diagonals bisect each other - Opposite sides are parallel - Two consecutive congruent sides
Naming Angles
- <=angle symbol - Refers ONLY to the angle you intend to name - Three letters to name an angle -- middle is vertex - Angles with angle bisectors cannot be named by a single letter since it isn't specific enough
Kites
- A four-sided polygon with exactly two pairs of consecutive congruent sides - The diagonal that goes in the middle of the kite divides it into two congruent triangles (the longer one) - The angles between the unequal sides are equal - The angles between the equal sides are bisected by the diagonal - The diagonal that goes in the middle of the kite divides it into two unequal isosceles triangles (the shorter one) - The longer diagonal divides each isosceles (which was the effect of being cut by the smaller diagonal) into two pairs of congruent right triangles
What are Proofs?
- A proof is a problem in which you are given some information and then asked to reach a certain valid conclusion in a logical way and according to certain rules - (below) the reasons - G: given - D: definitions - P: postulate (properties) - T: theorems
Trapezoids
- A quadrilateral with exactly one pair of parallel sides - The base of the trapezoids are the pair of opposite sides that are parallel -Named by its vertices - Use a small trapezoid figure to indicate it - The two same side interior angles are supplementary
Isosceles Trapezoid
- A trapezoid with both legs congruent - The bases of a trapezoid of an isosceles can never be equal since then that would be a parallelogram - The base angles of an isosceles trapezoid are congruent
Altitude's Location obtuse
- Acute Triangles: Interior of the triangle - Right Triangle: right angle to hypotenuse = interior; two legs=altitude (since they are already perpendicular) - Obtuse Triangle: Extend side opposite the angle and draw segment perpendicular to line drawn
Angles with Categories?
- Acute: 0° to 89° - Right: 90° - Obtuse: 91° to 179° - Straight: 180°
Eight Angles Two Measures
- Alternate Interior Angles, Corresponding Angles, and Alternate Exterior Angles are congruent - Same Side (Consecutive) Interior Angles are supplementary
Quadrilaterals are Parallelograms when...
- Are parallel - Are congruent - Are equal - Diagonals bisect each other - One pair of opposite sides are both equal and parallel (or)
Sum of Exterior Angles
- As n increases, the sum of interior angles goes up. But, as n increases, the sum of the exterior angles stays the same. - Always 360° (theorem)
Inequalities for One Triangle
- As the size of an angle increases, the sides of the angle move further apart - As the sides of an angle move further apart, the size of an angle increases
Trigonometry
- Based on facts that the value of the ratio of the lengths of two particular sides of a right triangle depends only on the size of the acute angle - The ratio determined by the measure of the acute angle and by the specific combination of sides that form the ratio
T2K about Trig Ratios
- Can divide an isosceles triangle into two congruent right triangles - Tan=sin/cos - sin^2+cos^2=1
Polygons
- Closed figures - Made up of straight line segments - Segments intersect at their endpoints - Segments intersect with exactly two other segments
Non-adjacent circles
- Common External Tangent: A tangent that does not cross a segment connecting the centers - Common Internal Tangent: A tangent that crosses a segment connecting the centers
Ratios
- Compare two numbers using division - Couple of ways to write them: 1/4, "one to four", "one is to four", 1:4 - Simplest form: 4/16 --> 1/4 - More than two terms (ex. 1:2:7)
Convex or Concave
- Convex: All edges are outer edges - Concave: At least one edge sinks inside
Parallel Lines
- Coplanar and do not intersect - Symbol = || - The same number of arrowheads on each of the two lines means that they are parallel (in the middle)
Relative Positions
- Corresponding Angles: A pair of angles in the same relative position but different line - Alternate Interior Angles: A pair of angles on opposite but interior sides. - Same Side Interior Angles: A pair of angles on the same but interior side. - Alternate Exterior Angles: A pair of angles on opposite but exterior sides.
Parallel Planes
- DO NOT INTERSECT - Be careful; Things that look parallel might intersect even though it may not look like it
Congruence
- Degree measure = equal - Every angle with the same degree measure is the same size - Congruence is the same as equal measure - When angles are marked with the same symbol, the angles are congruent
Proofs
- Direct proofs: given some information and asked to reach a certain conclusion by adding other information - Indirect proofs: given some information and asked to reach a certain conclusion by temporarily assuming the prove is negative and adding information until we contradict the given
Area of Rectangles
1) A rectangle is a quadrilateral with four right angles. 2) The perimeter of a rectangle with base b and height h is 2b + 2h. 3) The area of a rectangle with base b and height h is bh. 4) To find the diagonal of a rectangle, use the Pythagorean Theorem. 5) If it gives a ratio, find the square root of the quotient of the total area to the area of the ratio and multiply it to the base and height.
Area of Squares
1) A square is a polygon with four equal sides and four right angles. 2) If the side of a square is s units long, its perimeter is 4s. 3) If the side of a square is s units longs, its area is s^2.
Theorems about Circles
1) A tangent is perpendicular to a radius or diameter drawn to the point of tangency. 2) If two tangents to a circle are drawn from the same external point, the tangent segments are congruent. 3) If a radius is perpendicular at its outer endpoint to a line in the plane of a circle, then the line is tangent to the circle.
Theorems and Postulates about Similar Figures
1) AA~ (Angle Angle Similarity Postulate): If two triangles have two pairs of congruent angle, then the two triangles are similar. 2) SAS~ (Angle Side Angle Similarity Theorem): If two triangle have two pairs of sides which are in proportion and the included angles are congruent, then the two triangles are similar. 3) SSS~ (Side Side Side Similarity Theorem): If two triangle have three pairs of sides which are in proportion, then the two triangles are similar.
Proving Triangles' Similarity~
1) AA~: If two triangles have two pairs of congruent angles then the two triangles are similar 2) SAS~: If two triangles have two pairs of sides which are in proportion and the included angles are congruent, then the two triangles are similar 3)SSS~: If two triangles have three pairs of sides which are in proportion, then the two triangles are similar
Properties about Angles
1) Addition Property of Equality: If x=y and a=b, then x+a=y+b. 2) Subtraction Property of Equality: If x=y and a=a (Refl.), then x-a=y-a. 3) Substitution Property of Equality: If a=b then either a or b can be substituted for each other in any equation. 4) Division Property of Equality: If x=y and c=/0, then x/c=y/c. 5) Multiplication Property of Equality: If x=y, then cx=cy. 6) Symmetric Property of Equality: If x=y, then y=x. 7) Reflexive Property: m<D=m<D, AB=AB, x=x. 8) Transitive Property: If AB=CD and CD=EF, then AB=EF.
Working with Regular Polygons
1) All radiuses are equal. 2) The radii divide a polygon with n sides into n congruent isosceles triangles. 3) The radiuses meet in the center of a polygon. 4) The radii divide the 360° angle into n equal sides (360/n degrees).
Lateral Area, Total Area, and Volume of Spheres
1) Area: 4πr^2 2) Volume: 4/3πr^3
Prisms
1) Bases: congruent, polygons, and in parallel planes (two bases) 2) Named by shape of their bases 3) The sides of prisms that are not bases are lateral faces (right prisms=faces are rectangles) 4) A lateral edge is a line segment where two lateral faces met (they are lines) 5) Cubes have edges that are all equal, so their Base is s^2, their height is the side, and the volume is s^3.
Theorems and Postulates about Parallel Lines
1) CCP (postulate): Congruent corresponding angles means lines are parallel. 2) PCC (converse of 1): Lines are parallel if there are congruent corresponding angles. 3) CAIAP (theorem): Congruent alternate interior angles mean lines are parallel. 4) SSISP (theorem): Supplementary same side interior angles mean lines are parallel. 5) CAEAP (theorem): Congruent alternate exterior angles mean lines are parallel.
#) Type of angle: Location of vertex (relationship to arc measure)
1) Central Angle: Center (=) 2) Inscribed Angle: On the circle (1/2) 3) Chord/Tangent: On the circle (1/2) 4) Intersection: Inside the circle (Arc + Arc /2) 5) Secants: Outside the circle (Big Arc - Small Arc / 2) 6) Tangents: Outside the circle (360°)`
The Area of a Rhombus
1) Diagonals = d1 and d2 2) Diagonals divide rhombus into four right triangles. 3) (d1)(d2) / 2
Area of a Regular Polygon
1) Divide into n triangles (n=sides). 2) Focus on one triangle and use other concepts to make it into a right triangle. 3) Use trigonometry and other methods to find the lengths of the sides. 4) There are two ways to go from here: 1. (Area of one right triangles)(Number of right triangles in the figure) 2. 1/2(apothem)(perimeter)
Theorems and Postulates about Equilateral Triangles
1) Equilateral triangles are equiangular. 2) Equiangular triangles are equilateral. 3) Each angle of an equilateral triangle equals 60°.
Exterior Angles of Triangles
1) Extend one side of the triangle 2) The angle formed by the extended side and the side of the triangle crossing it is an exterior angle 3) As you move counter clockwise, keep going (repeat three times for the three vertices).
Theorems and Postulates for Inequalities
1) Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is larger than the measure of either of the two remote interior angles. 2) In a triangle, the sum of the lengths of the two smaller sides is larger than the length of the third side. 3) Within a single triangle, if one side is larger than a second side, then the angle opposite the first side is larger than the angle opposite the second side. (use when you have info. about sides) 4) Within a second triangle, if one angle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle. (use when you have info. about angles)
Theorems and Postulates about Angle Bisectors and Isosceles Triangles
1) If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. 2) If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. 3) ITT: If a triangle has two congruent sides, the angles opposite those sides are also congruent. 4) CITT: If a triangle has two congruent angles, the sides opposite those angles are also congruent.
Properties of Inequalities
1) If a>b and c>=d, then a+c>b+d. 2) If a>b and b>c, then a>c. 3) If a=b+c and b>0 and c>0, then a>b and a>c.
Logical statements
1) If-then statement: If p, then q. 2) Inverse: If not p, then not q. 3) Converse: If q, then p. 4) Contrapositive: If not q, then not p. - An if-then statement and its contrapositive are both true or both false - A converse and the inverse are whether both true or both false - If p->q and q->r, then p->r
T2K about Similarity
1) In Proportions: the ratios of the lengths of the corresponding sides are the same 2) Scale Factor: the reduced ratio of the corresponding sides 3) Order of Scale Factor: "small to big" or "big to small" 4) Order of Similarity: Four angle congruencies and an extended proportion ex: ABCD~FGHI - congruency: <A~=<F, <B~=<G, <C~=<H,<D~=<I - extended proportion: AB/FG, BC/GH, CD/HI, DA/IF
Cones
1) It comes to a point. 2) The base is a circle.
Lateral Area, Total Area, and Volume of Prisms
1) Lateral area (the area of the lateral faces): L.A.=ph (p=perimeter of base, h=height) or all the areas of the rectangles 2) Total area (lateral area + area of the 2 bases): T.A.=L.A.+2B 3) Volume: V=Bh
Lateral Area, Total Area, and Volume of Cones
1) Lateral area: πr[cursive]l (slant height) 2) Total area: πr[cursive]l+πr^2 3) Volume: 1/2πr^2h
Lateral Area, Total Area, and Volume of Pyramids
1) Lateral area: 1/2[cursive]pl or just multiply the area of one triangle to the number of faces 2) Total area: L.A.+1B 3) Volume: 1/3Bh
Lateral Area, Total Area, and Volume of Cylinders
1) Lateral area: 2πrh 2) Total area: 2πrh+2r^2 3) Volume:πr^2h
Inequality Proofs "Check List"
1) Look for isosceles triangles (opposite angles are equal). 2) Look for exterior angles, which help you put the angles in size order. 3) Combine what you have learned in Steps 1 and 2.
Trig Functions
1) Opposite: Side opposite from the angle 2) Hypotenuse: The longest side of a triangle 3) Adjacent: Right next to the angle 1) Sin: length of opposite leg/length of hypotenuse 2) Cos: length of adjacent leg/length of hypotenuse 3) Tan: length of opposite leg/length of adjacent leg
T2L4 in Proving Similarity
1) Parallel symbols (equal angles) 2) Intersecting Lines (VAT) 3) Common (shared) parts (Shared angles)
Terms of Transformations
1) Preimage: The original figure before any transformation 2) Image: The transformed figure 3) Isometry: The preimage and image are congruent 4) Rigid Motion: A motion that preserves both lengths and angles
Theorems and Postulates for Inequalities con.
1) SAS Ineq.: If two pairs of sides of two triangles are equal and the included angle of the first triangle is smaller than the included angle of the second triangle, then the third side of the first triangle is smaller than the third side of the second triangle. 2) SSS Ineq.: If two sides of one triangles are equal to two sides of another triangle and the third side of the first triangle is smaller than the third side of the second triangle, then the angle opposite the third side of the first triangle is smaller than the angle opposite the third side of the second triangle.
Which Theorem to Use -- SAS Ineq. or SSS Ineq.
1) SAS Ineq: Use if you know information about two pairs of sides and the included unequal angles since it tells us which of the sides opposite the unequal angles is larger and vice versa 2) SSS Ineq: Use if you know information about three pairs of sides, two pairs equal, one pair unequal since it tells us which of the angles opposite the unequal sides is larger and vice versa
Lengths, Area, and Volumes of Similar Solids
1) Scale Factor: The value of the common ratio of corresponding lengths 2) In proportion with one another When you have a ratio, switch it to... 1. Level 1: The ratio of all corresponding lengths of two objects equals their scale factor. 2. Level 2: The ratio of all corresponding areas of the two objects equals their scale factor squared. 3. Level 3: The ratio of the volumes of the two figures equals their scale factor cubed.
Sectors and Their Areas
1) Sectors is formed by two radii and an arc 2) The area is x/360(πr^2)
Planes Passing Through Spheres
1) Solid wooden ball 2) A plane cuts through the ball 3) Top of bigger piece is a circle, which is called the circle of intersection 4) The radius, the distance from the center of the circle to the center of the sphere, and the radius of the sphere will create a right triangle 5) You can use the Pythagorean Theorem to find the length of any of the above
ESSENTIALS!
1) The Pythagorean Theorem (use when you have a right triangle and the lengths of two sides are given) 2) The Law of Sines for Right Triangles (use when you have a right triangle and the measures of one side and either acute angle is given) note: sin of 90° is 1 3) Solving Isosceles Triangles (draw the altitude from the vertex angle to the base and use the Law of Sines to solve one of the two right triangles that form)
Theorems and Postulates about Trapezoids and Parallel Lines
1) The base angles of an isosceles trapezoid are congruent. 2) The length of the median of a trapezoid is equal to the average length of its bases. 3) The median of a trapezoid is parallel to the bases. 4) The segment connecting the midpoints of two sides of a triangles is half as long as the third side and parallel to it. 5) If three or more parallel lines cut off equal segments on a transversal, they are the same distance apart. If three or more parallel lines are the same distance apart, they cut off equal segments on a transversal.
Regular Pyramids
1) The base is a regular (equal sides and equal angles) polygons. 2) The faces are congruent, isosceles, and triangles. 3) The vertex is directly above the center of the base and therefore the altitude meets the base at its center. 4) The slant height of a pyramid is the height of a face. The symbol for the slant height is a lower case cursive L.
Theorems and Postulates about Triangles
1) The interior angles of a triangle total 180°. 2) Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. 3)SSS: If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. 4) SAS: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. 5) ASA: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.
Arc Length
1) The length of an arc is the linear measure of a part of the circumference. The measure of an arc means the degree measure, but the length means the linear measure. 2) Arc Length=x/360(2πr) 3) (above) This is because the arc is a degree over the total degrees in the circle (which is 360) 4) You can use this equation to find the arc length or the arc measure.
Area of Parallelograms
1) The length of the altitude is the height. 2) The area is (base)*(height). 3) You can use the area of a parallelogram when only an angle and a length is given. Draw the altitude (which forms a right angle with the hypotenuse) and use trigonometry to find the opposite. 4) Remember: A base can be a height and a height can be the base, it is just the height to that specific base.
Facts about Kites
1) The opposite sides are never equal. 2) The angles between the equal sides are never equal. 3) The diagonals are never equal. 4) The diagonals are perpendicular (thus forming right triangles). 5) The longer diagonal bisects the shorter diagonal.
Circle Segments
1) The segment is the area formed by a chord and an arc. 2) To find the area of a segment, find the area of the sector, the triangle, and then subtract them.
Congruent Arcs
1) The two arcs must have the same measure. 2) The two arcs must be in the same or in congruent circles.
Radical Rules
1) There are no perfect square factors left under the radical sign. 2) There are no radicals in the denominator. 3) There are no fractions under the radical sign.
T2K about Circles
1) They are made up of all of the points in a plane that are equally far away from a given point called the center. 2) The distance from the center to the circle is called the radius (All radii of a circle are equal). 3) Circles are named by their centers which are points, and points are named by a single capital letter. 4) They are associated with a dot and a circle around it. 5) These are terms associated with circles: - Chord: A line segment whose endpoints are both on the circle - Diameter: A chord that passes through the center of a circle (the longest chord in the circle) - Secant: A line that includes a chord of a circle - Tangent: A line which passes through only one point of a circle, or one of its rays or segments which includes that point - Point of Tangent The single point shared by a circle and a tangent 6) All circles have the same shape and therefore all circles are similar. 7) Congruent circles have congruent radii. 8) Concentric circles are circles having the same center.
Pyramids
1) They come to a point called a vertex. 2) The sides of the pyramid are triangles called lateral faces. 3) Each edge where two lateral faces meet is called a lateral edge. 4) The altitude is the perpendicular segment from the vertex to the base. 5) The length of the altitude is the height of the pyramid.
Area of Triangles
1) Triangles are 1/2 of a parallelogram 2) The area would be (base)*(height to that base) /2 3) The height is the length of the altitude 4) For right triangles, it would be (Leg)*(Leg) /2. 5) For isosceles triangles, it can be divided into two congruent right triangles (look at 4 and then multiply by two). 6) You can use trigonometry for it if you only have the angle measure and a side length.
What makes an angle?
1) Two rays 2) Share a common endpoint 3) Common endpoint = vertex of angles 4) Named by a single capital letter 5) Includes infinite "interior region" between rays
Theorems and Postulates about Angles
1) VAT (Vertical Angle Theorem): Vertical angles are congruent. 2) Angle Addition Postulate: If point K lies in the interior region of <JOL, then m<JOK + m<KOL = m<JOL. 3) Angle Bisector Theorem: If BD is the bisector of <ABC, then <ABD~=½<ABC and <DBC~=½<ABC. 4) If <A~=<B and <B~=<C then <A~=<C. 5) Complements of the same or congruent angles 6) Supplements of the same or congruent angles are congruent. 7) If two lines form congruent adjacent angles, the lines are perpendicular.
Triangles: What you should know
1) Vertical angles are congruent. 2) A straight angles measures 180°. 3) The three angles of a triangle add up to 180°.
Theorems and Postulates about Right Angles
1) When the altitude is drawn from the right angle to the hypotenuse of a right triangle, the two triangles that are formed are similar to the original triangle and similar to each other. 2) In a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean between the two segments of the hypotenuse. 3) In a right triangle, when you draw the altitude from the right angle to the hypotenuse, the length of each leg is the geometric mean between the entire hypotenuse and the part of the hypotenuse nearest the leg.
The Area of a Trapezoid
1/2(h)(b1+b2) or (h)(median) note: b1 and b2 = bases note: altitudes divide an isosceles trapezoid into two congruent right triangles and a rectangle
Theorems about Circles con.
4) Congruent arcs have congruent central angles. 5) Congruent central angles have congruent arcs. 6) In the same or in congruent circles congruent arcs have congruent chords. 7) In the same or in congruent circles congruent chords have congruent arcs. 8) In the same or congruent circles, congruent chords are equally distant from the center of the circle.
Theorems and Postulates about Triangles con.
6) Hypotenuse Leg Theorem: If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of a second right triangle, the two triangles are congruent. 7) CPCT: Corresponding Parts of Congruent Triangles are Congruent. 8) Every point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. 9) Any point that is equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.
Theorems about Circles con2.
9) In the same or congruent circles, chords which are equally distant from the center are congruent. 10) A diameter that is perpendicular to a chord bisects the chord and its arc. 11) Inscribed Angle Theorem: An arc equals two times its inscribing angle and an inscribed angle is 1/2 the arc. 12) The angle formed by a chord and a tangent equals one half the measure of the arc of the chord.
Remote Interior Angles
The two interior angles away from the exterior angles are remote interior angles
Transversal
A line that crosses two coplanar lines at two different points
Regular Polygon
A polygon with equal sides and equal angles
General Triangles
Any kind of triangle, scalene, isosceles, equilateral, acute, right, obtuse, etc. - six parts (three angles, three sides)
Geometric Probability
Area of win ------------- Area of whole figure - Works for length and time - Multiply types of categories - (five tops, three pairs of shorts and two jackets = 30 different outfits)
Central Angles
The vertex of an angle is placed at the center of a circle and the sides of the angle intercept the circle and determine an arc
Inscribed and Circumscribed
Inscribed Angles/Polygons--Circumscribing Circles: A polygon/angle is inscribed in a circle if every vertex of the polygon/angle lies on the circle. The circle is said to circumscribe the polygon. Inscribed circles--Circumscribing Polygons A circle is inscribed in a polygon if every side of the polygon is tangent to the circle. The polygon is said to circumscribe the circle.
Complementary and Supplementary
Complementary: - Two angles whose measures total 90° - These angles may or may not be adjacent Supplementary: - Two angles whose measures total 180° - These angles may or may not be adjacent
The Two Tangent Case
If the sides of the angle are two tangents, the connection between the angle and the arcs is the same, but the arcs have a special connections with each other (measures 360°)
Semicircles
If two points are at the outer endpoints of a diameter, the points divide the circle into two semicircles, which measure 180°
Angles with Vertices Outside the Circle
If... -Vertex is outside - Sides are two secants, secant and a tangent, or two tangents Then... <=big arc - small arc /2
Exterior Angles of Polygons
Extend the side of the polygon - For all polygons: sides = interior angles = exterior angles - For all regular polygons: sides are equal, interior angles are equal, and exterior angles are equal
The Pythagorean Theorem
If a triangle is a right triangle, then (Leg 1)^2 + (Leg 2)^2 = (Hypotenuse)^2. 1) The length of the hypotenuse always sits all by itself on one side of the equation. 2) The hypotenuse is opposite the right angle and is always the longest side. 3) This theorem only works for right triangles. 4) You must square each length individually. (The Distance Formula uses the Pythagorean Theorem to find the distance between two points)
Theorem: Obtuse or Acute?
Obtuse: (Side 1)^2 + (Side 2)^2 < (Longest Side)^2 Acute: (Side 1)^2 + (Side 2)^2 > (Longest Side)^2
Theorems and Postulates about Quadrilaterals
Parallelograms: 1) Both pairs of opposite sides and angles are equal. 2) The diagonals divide each other in half. Quadrilaterals: 1) A quadrilateral with two pairs of equal opposite sides is a parallelogram. 2) If both diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. 3) If one pair of opposite sides are congruent and parallel, it is a parallelogram.
Circumference
Perimeter for circles Formula: C=2πr or C=πd
Naming Polygons
Polygons are named by their consecutive vertices, which are points
Finding Trig Ratios with Scientific Calculators
Press the key of the particular trig ratio you wish to find and then enter the size of the angle and then press the equal sign (To find the angle: Press the 2nd key, then the key for the trig ratio you are going to enter, then enter the measure of the trig ratio and then press equal)
Theorems and Postulates about Quadrilaterals con.
Rectangles: 1) The diagonals of a rectangle are equal. 2) In a right triangle the midpoint of the hypotenuse is equidistant from each vertex. Rhombuses: 1) The diagonals of a rhombus are perpendicular. 2) The angles of a rhombus are bisected by its diagonals.
Right or Oblique
Right=perpendicular to the ground Oblique=not perpendicular to the ground, slanted
Learning Trigonometric Functions
Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse Tangent Opposite Adjacent
Heron's Formula
The area of a triangle with sides a, b, and c is: A= √S(S-a)(S-b)(S-c)
Regular: Equal sides and equal angles
The center of a circumscribing circle is the center of a polygon. The apothem is a perpendicular segment from the center of a regular polygon to a side. The radii and the apothems divide a regular polygon into 2n congruent right triangles.
Spheres
The collection of points in space that are equally far away from a given point called its center
Midsegment
The median of a triangle in terms of trapezoid definition
Segment Divided Proportionally
Two or more segments that are divided so that the ratio of the lengths of their parts are the same
How to Find the Total Degrees
Use the equal (n-2)(180) n=number of sides
Segments and Circles
When two chords of a circle intersect, each chord is divided into two parts.
Law of Sines
sin<A/a = sin<b/b = sin<c/c (Use to solve a triangle when you have three pieces of information and at least two pieces are a pair, that is, an angle and the side opposite that angles)
Radical Talk
√2: -Radical 2 -Rad 2 -Root 2 -The square root of 2