Math B - Geometry
Congruent Triangles
· Can be proven by
o SSS – side side side
o SAS – side angle side
o ASA – angle side angle
o AAS – angle angle side (because if we know 2 angles, we can figure out the third, making ASA) hy-acute
· A bisector divided something into 2 equal parts
o A=B
· CPCTC – Corresponding Parts of Congruent Triangles are Congruent
· Perpendicular bisector forms 2 equal 90* angles
o <1, <2 are right angles
· All right angles are congruent
· A midpoint cuts the line in half
· Identity/Reflexive Postulate – Everything is equal to itself
· If 2 sides of a triangle are congruent, the angles opposite them are congruent.
· If 2 angles of a triangle are congruent, the sides opposite them are congruent.
· Subtraction Postulate – equals are subtracted from equals, the results are equal
· Supplements of equal angles are equal
· All points on the perpendicular bisector of a line segment are equidistant from the endpoints of the line segment
· From a point outside a line, a line can be drawn perpendicular to the given line
o BD is perpendicular to AC --- drawn perpendicular line
· All points on a perpendicular bisector are equidistant from the endpoint
· All points on the angle bisector are equidistant to the rays of the angle
· (An exterior angle of a triangle is equal to the sum of the 2 remote interior angles)
Indirect proofs
· Given
· Write opposite of what you want to prove – “assuming the opposite of what I want to prove”
· Try to prove something you know is false
· What you wanted to prove – “Because my assumption led to a contradiction, my assumption must be false”
Quadrilaterals – not complete. Look at descriptions
· If alternate interior angles are congruent when lines are cut by a transversal
· If parallel lines are cut by a transversal, the alternate interior angles are equal
· If one pair of opposite sides are parallel and congruent, the quadrilateral is a parallelogram
o Proving a quadrilateral to be a parallelogram
· Vertical angles are equal
· Opposite sides of a parallelogram are equal
· If both pairs of opposite sides of a quadrilateral are congruent – parallel
· Opposite sides of a rectangle are equal
· If 2 things are equal to the same thing, then they must be equal to each other.
· If one pair of opposite sides are parallel and equal, the quadrilateral is a parallelogram
· Whole is greater than any of its parts
Similar Triangles
· Can be proven by
o 2 equal angles
§ A.A. theorum
· Proofs
o Corresponding sides of similar triangles are in proportion
§ YT/YR = XY/ZY
o In a proportion, the product of the means equals the product of the extremes
o When 2 adjacent congruent angles are resting on a straight line, they are both right angles
· Short answers
o The line joining the midpoint of 2 sides of a triangle is parallel to the 3rd side and equal to half its length
o When an altitude is drawn to the hypotenuse of a right triangle, the altitude is the mean proportion between the 2 segments of the hypotenuse
o When an altitude is drawn to the hypotenuse of a right triangle, the leg is the mean proportional between the whole hypotenuse and the segment of the hypotenuse near that leg
Circles
Chord – line segment that joins two points on a circle
Secant – line that joins 2 points but continues outside the circle
Tangent – touches circle in one point
Arcs
· Central angles – vertex is the center
o Equal to the number of degrees of the arc in front of it
· Inscribed angle – vertex of angle is on circumference
o ˝ its arc
· Inside angle – vertex is inside the circle
o ˝ the sum of the arcs in front of it and in back
· Outside angle – made up of two secants
o ˝ the difference of the 2 arcs in front of it
· Tan Chord
o ˝ arc in front of it
Length
· When two chords intersect in a circle, the product of their respective segments are equal
· Whole * Outside = Whole * Outside
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