Download Review: key postulates and theorems (6.0

Review: key postulates and theorems (6.0-6.13)
Numbers in parentheses refer to textbook sections. Also note that the textbook has a similar
list covering the entire book on pp. 1008-1013; the Chapter 6 section of that list might be
useful now. This is not necessarily a comprehensive list of everything we have covered this
unit.
Postulates
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All the basic facts and rules of algebra may be used.
Through two points, there is exactly one line..
A whole equals the sum of its parts.
Triangle Congruence Postulates: SSS, SAS, ASA. (6.00)
Parallel Postulate: Given a point and a line, there is exactly one line through the
given point parallel to the given line. (6.07)
CPCTC – Corresponding parts of congruent triangles are congruent.
Theorems
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Vertical Angle Theorem: Vertical angles are congruent. (6.05)
AIP Theorem: If two lines form congruent alternate interior angles with a transversal,
then the two lines are parallel. Also similar facts about other types of angles such as
corresponding angles, consecutive angles, etc. (6.06)
PAI Theorem: If two lines are parallel, then alternate interior angles formed with a
transversal are congruent. Also similar facts about other types of angles such as
corresponding angles, consecutive angles, etc. (6.07)
Triangle Angle Sum Theorem: The angle sum of any triangle is 180°. (6.07)
Unique Perpendicular Theorem: Given a point and a line, there is exactly one line
through the given point perpendicular to the given line. (6.07 exercise 4)
AAS Triangle Congruence (6.07 exercise 11)
Exterior Angle Theorem (first version): An exterior angle of a triangle has greater
measure than either of the remote interior angles. (6.04 exercise 5)
Exterior Angle Theorem (second version): The measure of an exterior angle of a
triangle is the sum of the measures of the two remote interior angles. (6.07 exercise
12)
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a line
segment then it is equidistant from the endpoints of the line segment. Also the
converse of this theorem. (6.13)
Isosceles Triangle Theorem: If a triangle has two congruent sides, then their opposite
angles are congruent. Also the converse of this theorem. (6.13)
Hypotenuse-Leg (HL) Triangle Congruence (6.13 exercise 13)
If two angles are both supplementary and congruent, then they are right angles.
If two angles are complementary to the same angle, then they are congruent.
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If two angles are supplementary to the same angle, then they are congruent.
Halves of congruent segments (or angles) are congruent.
Quadrilaterals Quiz Review
This quiz will cover Investigation 6D in our textbook. This includes section 6.14-6.18.
Here is a summary of some key theorems for proving special types of quadrilaterals.
However, this is not a comprehensive list of all theorems proved in this unit.
Ways to prove a parallelogram
• If a quadrilateral has both pairs of opposite sides congruent, then the quadrilateral
is a parallelogram.
• If a quadrilateral has each pair of consecutive angles supplementary, then the
quadrilateral is a parallelogram.
• If a quadrilateral has both pairs of opposite angles congruent, then the
quadrilateral is a parallelogram.
• If each diagonal of a quadrilateral divides it into two congruent triangles, then the
quadrilateral is a parallelogram.
• If a quadrilateral has diagonals that bisect each other, then the quadrilateral is a
parallelogram.
• If a quadrilateral has one pair of opposite sides congruent and parallel, then the
quadrilateral is a parallelogram.
Ways to prove a rectangle
• If a parallelogram has at least one right angle, then the parallelogram is a
rectangle.
• If a parallelogram has congruent diagonals, then the parallelogram is a rectangle.
Ways to prove a rhombus
• If a parallelogram has two adjacent sides congruent, then the parallelogram is a
rhombus.
• If a quadrilateral has four congruent sides, then the quadrilateral is a rhombus.
• If a parallelogram has perpendicular diagonals, then the parallelogram is a
rhombus.
• If one diagonal of a parallelogram bisects two opposite angles, then the
parallelogram is a rhombus.
• If both diagonals of a quadrilateral bisect the vertex angles, then the quadrilateral
is a rhombus.
Ways to prove a square
• If a quadrilateral is both a rectangle and a rhombus, then the quadrilateral is a
square.
Circles – Summary of Key Formulas and Theorems
Below is a summary of some of the key theorems from Chapter 8. This list is not
necessarily a comprehensive list of all the theorems we’ve seen in Chapter 8.
Key Terms:
apothem, arc, arc length, central angle, chord, diameter, minor arc, major arc, arc
measure, inscribed angle, circumscribed circle, circumcenter, inscribed circle, incenter,
cyclic quadrilateral, secant, secant angle, tangent
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Area of a regular polygon: A = Pa Area of a circle: A = Cr or A = πr 2
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θ
Circumference of a circle: C = 2πr
Arc Length: L = 2πr •
360
θ
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Sector Area: A =€πr •
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360
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Theorems:
• If two central angles are congruent, then their intercepted arcs are congruent (and
€ converse)
• Two chords are congruent if and only if their corresponding arcs are congruent
• If a line goes through the center of a circle and bisects a chord, then the line is
perpendicular to the chord.
• If a line goes through the center of a circle and is perpendicular to a chord, then
the line bisects the chord.
• If a line is the perpendicular bisector of a chord, then it passes through the center
of the circle.
• The measure of an inscribed angle is equal to half the measure of its intercepted
arc (Inscribed Angle Theorem)
o Inscribed angles are congruent if and only if they intercept the same arc or
congruent arcs
o Any triangle inscribed in a semicircle is a right triangle.
• In a cyclic quadrilateral, opposite angles are supplementary.
• If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of
tangency.
• A secant angle with the vertex inside a circle is equal in measure to half of the
sum of the arcs it intercepts (Secant Angle Theorem)
• A secant angle with the vertex outside a circle is equal in measure to half of the
difference of the arcs it intercepts (Secant Angle Theorem)
o Note: This theorem is true is one or both of the lines are tangent to the
circle.
• Two segments through a common point and tangent to the same circle are
congruent (Two Tangent Theorem)