Download Ways to Prove Angles Congruent Ways to Prove Angles

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Ways to Prove
Angles Congruent
• Vertical angles are congruent.
• If two angles are supplements of congruent angles (or the same
angle) the two angles are congruent.
• If two angles are complements of congruent angles (or of the same
angle) the two angles are congruent.
• If two lines are perpendicular they form congruent adjacent angles.
• If parallel lines are cut by a transversal, the corresponding angles are
congruent.
• If parallel lines are cut by a transversal, the alternate interior angles
are congruent.
• If an angle is bisected, it divides it into two congruent angles.
• If two angles are equal in measure, then they are congruent.
(Definition of Congruent Angles)
• If two angles in one triangle are congruent to two angles in another
triangle, the third angles are congruent.
• The exterior angle of a triangle equals the sum of the two remote
interior angles.
Ways to Prove Angles
Complementary
• If the exterior sides of 2 adjacent angles are perpendicular the
angles are complementary.
• If the sum of the measure of two angles is 90 degrees, the angles
are complementary (Definition of complementary angles)
• The acute angles of a right triangle are complementary.
Ways to Prove Angles
Supplemenatry
• If parallel lines are cut by a transversal, the interior angles on the
same side of the transversal are supplementary.
• If two angles add up to 180 degrees, then they are supplementary
(Defintion of Supplementary Angles)
Angle
AngleBisector
AdditionTheorem
Axiom
Ways to Prove Lines
Perpendicular
Segment Addition Axiom
Midpoint Theorem
B is isinthe
thebisector
interior of angle ABC:
CDA then
the measure
of angle
• If BX
The measure
of angle
ABX =
CDB
the measure
of angle
BDA
the measure
of angle
1/2
the+ measure
of angle
ABC;
The=Measure
of angle
SBCCDA
= 1/2
the measure of angle ABC
• If a line is perpendicular to one of two parallel lines, it is
perpendicular to the other one as well.
• If two lines form right angles, then they are perpindicular.
(Definition of perpendicular lines)
• If B is between A and C then AB + BC = AC (whole = sum of its
parts)
• If M the midpoint of AB: AM = 1/2 AB; BM = 1/2 AB
Ways to Prove
Lines Parallel
Lines, Points, and
Planes
• If two lines are perpendicular to the same line, they are parallel
to each other.
• If two lines are parallel to a third line, then the two lines are
parallel to each other.
• If two lines are cut by a transversal and corresponding angles are
congruent, then the lines are parrallel.
• If two lines are cut by a transversal and alternate interior angles
are congruent, then the lines are parallel.
• If two lines are cut by a transversal and the same-side interior
angles are supplementary, then the lines are parallel.
• In a plane two lines perpendicular to the same line are parallel.
• If 2 lines intersect, they intersect in exactly one point.
• Though a line and a point not in the line there is exactly one
plane.
• If two lines intersect exactly one point contains the lines.
• A line contains at least two points.
• A plane contains at least three points not all in one line.
• Space contains at least four points not all in one plane.
• Through any two point there is exactly on line.
• Through any three point there is at least on plane, and
through any three non-collinear points there is exactly on
plane.
• If two points are in a plane, then the line that contains the
points in in that plane.
• If two planes intersect, then their intersection is a line.
• Through a point outside a line, there is exactly one line
parallel to the given line.
Definitions
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Line Segment:
 A part of a line with two end points.
Midpoint of a Segment:
 Divides the segment into two equal parts.
Bisector of a Segment:
 Intersects the segment at its midpoint; could be a line or a segment.
Congruent Segments:
 Two segments that have the same measure.
Perpendicular Lines:
 Two lines that intersect to form right angles.
Parallel Lines:
 Two lines on the same plane that never intersect.
Ray:
 A line that only goes in one direction.
Opposite Rays:
 Two rays that have the same endpoint that form a straight line.
Angle:
 The union of two rays with a common endpoint (vertex).
Adjacent Angles:
 Two angles that share a ray, share a vertex, and do not overlap.
Obtuse Angle:
 An angle was a measure greater than 90 degrees but less than 180 degrees.
Acute Angle:
 An angle with a measure between 0 and 90 degrees.
Right Angle:
 An angle with a measure of 90 degrees.
Bisector of an Angle:
 A ray, line, or sector that cuts an angle into two equal halves.
Vertical Angles:
 Two non adjacent angles formed by two intersecting lines.
Supplementary Angles:
 Angles that add up to 180 degrees when combined.
Complimentary Angles:
 Angles that add up to 90 degrees when combined.
Congruent Angles:
 Two angles that have the same measure.
Collinear:
 Two points on the same line.
Coplanar:
 Two points on the same plane.
 Isosceles triangle – a triangle with 2 sides congruent
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Scalene triangle – a triangle with no sides the same length
Equilateral triangle: a triangle with all sides congruent
Right triangle- a triangle with one right angle
Obtuse triangle- a triangle with one obtuse angle
Acute triangle – a triangle with all acute angles.
Properties of Equality
Subtraction
Addition
Multiplication
Division
Substituion
Reflexive
Symmetric:
Transitive
• If a = b and c = d then a - c = b - d
• If a = b and c = d then a + c = b + d
• If a = b then ac = bc
• If a = b and c ≠ d then a/c = b/c
• If a = b then you can substitute a for b or b for a in an
equation
• a=a
• If a = b then b = a