# Postulates, Theorems, and Corollaries

Electronics - Devices

Oct 8, 2013 (4 years and 8 months ago)

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S82 Postulates, Theorems, and Corollaries
Chapter 1
Post. 1-1-1 Through any two points there is
exactly one line. (p. 7)
Post. 1-1-2 Through any three noncollinear points
there is exactly one plane containing them. (p. 7)
Post. 1-1-3 If two points lie in a plane, then the
line containing those points lies in the plane.
(p. 7)
Post. 1-1-4 If two lines intersect, then they
intersect in exactly one point. (p. 8)
Post. 1-1-5 If two planes intersect, then they
intersect in exactly one line. (p. 8)
Post. 1-2-1
Ruler Postulate
The points on a line
can be put into a one-to-one correspondence
with the real numbers. (Ruler Post.; p. 13)
Post. 1-2-2
If B is
between A and C, then AB +BC =AC. (Seg. Add.
Post.; p. 14)
Post. 1-3-1
Protractor Postulate
Given " \$% AB and
a point O on " \$% AB, all rays that can be drawn from
O can be put into a one-to-one correspondence
with the real numbers from 0 to 180. (Protractor
Post.; p. 20)
Post. 1-3-2
If S is in
the interior of ∠PQR, then m∠PQS + m∠SQR =
Thm. 1-6-1
Pythagorean Theorem
In a right
triangle, the sum of the squares of the lengths of
the legs is equal to the square of the length of the
hypotenuse. (Pyth. Thm.; p. 45)
Chapter 2
Thm. 2-6-1
Linear Pair Theorem
If two angles
form a linear pair, then they are supplementary. (Lin. Pair Thm.; p. 110)
Thm. 2-6-2
Congruent Supplements Theorem
If two angles are supplementary to the same
angle (or to two congruent angles), then the two
angles are congruent. ('Supps. Thm.; p. 111)
Thm. 2-6-3
Right Angle Congruence Theorem
All right angles are congruent. (Rt.∠'Thm.;
p. 112)
Thm. 2-6-4
Congruent Complements Theorem
If two angles are complementary to the same
angle (or to two congruent angles), then the two
angles are congruent. ('Comps. Thm.; p. 112)
Thm. 2-7-1
Common Segments Theorem
Given
collinear points A,B,C, and D arranged as
shown, if
((
AB'
((
CD, then
((
AC '
((
BD. (Common
Segs. Thm.; p. 118)

Thm. 2-7-2
Vertical Angles Theorem
Vertical
angles are congruent. (Vert.)Thm.; p. 120)
Thm. 2-7-3 If two congruent angles are
supplementary, then each angle is a right angle. (') supp.→rt.); p. 120)
Chapter 3
Post. 3-2-1
Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then
the pairs of corresponding angles are congruent. (Corr.)Post.; p. 155)
Thm. 3-2-2
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of alternate interior angles are
congruent. (Alt. Int. ) Thm.; p. 156)
Thm. 3-2-3
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then
the two pairs of alternate exterior angles are
congruent. (Alt. Ext.) Thm.; p. 156)
Thm. 3-2-4
Same-Side Interior Angles Theorem
If two parallel lines are cut by a transversal, then
the two pairs of same-side interior angles are
supplementary. (Same-Side Int. ) Thm.; p. 156)
Post. 3-3-1
Converse of the Corresponding
Angles Postulate
If two coplanar lines are cut
by a transversal so that a pair of corresponding
angles are congruent, then the two lines are
parallel. (Conv. of Corr. ) Post.; p. 162)
Post. 3-3-2
Parallel Postulate
Through a point P
not on line ℓ, there is exactly one line parallel to
ℓ. (Parallel Post.; p. 163)
Thm. 3-3-3
Converse of the Alternate Interior
Angles Theorem
If two coplanar lines are
cut by a transversal so that a pair of alternate
interior angles are congruent, then the two lines
are parallel. (Conv. of Alt. Int. ) Thm.; p. 163)
Thm. 3-3-4
Converse of the Alternate Exterior
Angles Theorem
If two coplanar lines are
cut by a transversal so that a pair of alternate
exterior angles are congruent, then the two lines
are parallel. (Conv. of Alt. Ext. ) Thm.; p. 163)
Postulates, Theorems, and Corollaries
S82
S86 Postulates, Theorems, and Corollaries
Chapter 8
Thm. 8-1-1 The altitude to the hypotenuse of a
right triangle forms two triangles that are similar
to each other and to the original triangle. (p. 518)
Cor. 8-1-2
Geometric Means Corollary
The
length of the altitude to the hypotenuse of
a right triangle is the geometric mean of the
lengths of the two segments of the hypotenuse.
(p. 519)
Cor. 8-1-3
Geometric Means Corollary
The
length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
(p. 519)
Thm. 8-5-1
The Law of Sines
For any △ABC
with side lengths a,b, and c,
sinA____
a
=
sinB____
b
=
sinC____
c
.
(p. 552)
Thm. 8-5-2
The Law of Cosines
For any △ABC
with sides a,b, and c,a
2
=b
2
+c
2
- 2bccosA,
b
2
=a
2
+c
2
- 2accosB, and c
2
=a
2
+b
2
-
2abcosC. (p. 553)
Chapter 9
Post. 9-1-1
The area
of a region is equal to the sum of the areas of its
nonoverlapping parts. (Area Add. Post.; p. 589)
Chapter 11
Thm. 11-1-1 If a line is tangent to a circle, then
it is perpendicular to the radius drawn to the
point of tangency. (line tangent to ⊙→ line ⊥ to
Thm. 11-1-2 If a line is perpendicular to a radius
of a circle at a point on the circle, then the line
is tangent to the circle. (line ⊥ to radius →line
tangent to ⊙; p. 748)
Thm. 11-1-3 If two segments are tangent to a circle
from the same external point, then the segments
are congruent. (2 segs. tangent to ⊙ from same
ext. pt. → segs. '; p. 749)
Post. 11-2-1
The
measure of an arc formed by two adjacent arcs
is the sum of the measures of the two arcs. (Arc
Thm. 11-2-2 In a circle or congruent circles:
(1) congruent central angles have congruent
chords, (2) congruent chords have congruent
arcs, and (3) congruent arcs have congruent
central angles. (' arcs have ' central ) have '
chords; p. 757)
Thm. 11-2-3 In a circle, if a radius (or diameter)
is perpendicular to a chord, then it bisects
the chord and its arc. (Diam.⊥ chord → diam.
bisects chord and arc; p. 759)
Thm. 11-2-4 In a circle, the perpendicular bisector
of a chord is a radius (or diameter). (⊥ bisector
of chord is diam.; p. 759)
Thm. 11-4-1
Inscribed Angle Theorem
The
measure of an inscribed angle is half the
measure of its intercepted arc. (Inscribed ∠ Thm.;
p. 772)
Cor. 11-4-2 If inscribed angles of a circle intercept
the same arc or are subtended by the same
chord or arc, then the angles are congruent.
(p. 773)
Thm. 11-4-3 An inscribed angle subtends a
semicircle if and only if the angle is a right angle.
(p. 774)
Thm. 11-4-4 If a quadrilateral is inscribed
in a circle, then its opposite angles are
supplementary. (Quad. inscribed in circle → opp.
) supp.; p. 775)
Thm. 11-5-1 If a tangent and a secant (or chord)
intersect on a circle at the point of tangency,
then the measure of the angle formed is half the
measure of its intercepted arc. (p. 782)
Thm. 11-5-2 If two secants or chords intersect in
the interior of a circle, then the measure of each
angle formed is half the sum of the measures of
its intercepted arcs. (p. 783)
Thm. 11-5-3 If a tangent and a secant, two
tangents, or two secants intersect in the exterior
of a circle, then the measure of the angle formed
is half the difference of the measures of its
intercepted arcs. (p. 784)
Thm. 11-6-1
Chord-Chord Product Theorem
If two chords intersect in the interior of a circle,
then the products of the lengths of the segments
of the chords are equal. (p. 792)
Thm. 11-6-2
Secant-Secant Product Theorem
If two secants intersect in the exterior of a
circle, then the product of the lengths of one
secant segment and its external segment equals
the product of the lengths of the other secant
segment and its external segment. (whole •
outside = whole • outside; p. 793)
Thm. 11-6-3
Secant-Tangent Product Theorem
If a secant and a tangent intersect in the exterior
of a circle, then the product of the lengths of the
secant segment and its external segment equals
the length of the tangent segment squared. (whole • outside =tangent
2
; p. 794)
S86
Thm. 11-7-1
Equation of a Circle
The equation
of a circle with center (h,k) and radius r
is (x -h)
2
+ (y -k)
2
=r
2
. (p. 799)
Chapter 12
Thm. 12-4-1 A composition of two isometries is an
isometry. (p. 848)
Thm. 12-4-2 The composition of two reflections
across two parallel lines is equivalent to
a translation. The translation vector is
perpendicular to the lines. The length of the
translation vector is twice the distance between
the lines. The composition of two reflections
across two intersecting lines is equivalent
to a rotation. The center of rotation is the
intersection of the lines. The angle of rotation
is twice the measure of the angle formed by the
lines. (p. 849)
Thm. 12-4-3 Any translation or rotation is
equivalent to a composition of two reflections.
(p. 850)
Angle Bisector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 23
Center of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . p. 774
Centroid of a Triangle . . . . . . . . . . . . . . . . . . . . . p. 314
Circle Circumscribed
About a Triangle . . . . . . . . . . . . . . . . pp. 313, 778
Circle Inscribed in
a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 313
Circle Through Three
Noncollinear Points . . . . . . . . . . . . . . . . . . . p. 763
Circumcenter of
a Triangle . . . . . . . . . . . . . . . . . . . . . . pp. 307, 313
Congruent Angles . . . . . . . . . . . . . . . . . . . . . . . . . .p. 22
Congruent Segments. . . . . . . . . . . . . . . . . . . . . . . .p. 14
Congruent Triangles
Using ASA. . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 253
Congruent Triangles
Using SAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 243
Congruent Triangles
Using SSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 248
Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . pp. 872, 878
Equilateral Triangle . . . . . . . . . . . . . . . . . . . . . . . p. 220
Incenter of a Triangle . . . . . . . . . . . . . . . . . . . . . p. 313
Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . p. 363
Kite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 435
Midpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 16
Midsegment of a Triangle. . . . . . . . . . . . . . . . . . p. 327
Orthocenter of a Triangle. . . . . . . . . . . . . . . . . . p. 320
Parallel Lines. . . . . . . . . . . . . . . pp. 163, 170, 171, 179
Parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 404
Perpendicular Bisector
of a Segment . . . . . . . . . . . . . . . . . . . . . . . . . p. 172
Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . p. 179
Rectangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 424
Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . pp. 824, 829
Regular Decagon . . . . . . . . . . . . . . . . . . . . . . . . . p. 381
Regular Dodecagon. . . . . . . . . . . . . . . . . . . . . . . p. 380
Regular Hexagon . . . . . . . . . . . . . . . . . . . . . . . . . p. 380
Regular Octagon. . . . . . . . . . . . . . . . . . . . . . . . . . p. 380
Regular Pentagon. . . . . . . . . . . . . . . . . . . . . . . . . p. 381
Rhombus . . . . . . . . . . . . . . . . . . . . . . . . . . pp. 415, 424
Right Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 258
Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . pp. 839, 844
Segment Bisector. . . . . . . . . . . . . . . . . . . . . . . . . . .p. 16
Square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . pp. 380, 424
Tangent to a Circle
at a Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 748
Tangent to a Circle from
an Exterior Point. . . . . . . . . . . . . . . . . . . . . . p. 779
Translation . . . . . . . . . . . . . . . . . . . . . . . . pp. 831, 836
Triangle Circumscribed
About a Circle . . . . . . . . . . . . . . . . . . . . . . . . p. 779
Trisecting a Segment. . . . . . . . . . . . . . . . . . . . . . p. 487
Constructions
Constructions S87
S87
pages S86–S87 Skills Bank 919
919