Geometry – Theorems

January 7, 2008 at 12:26 am 32 comments

Theorems
2.1 Properties of Segment Congruence Segment
congruence is reflexive, symmetric, and transitive.

2.2 Properties of Angle Congruence Angle
congruence is reflexive, symmetric, and transitive.
Reflexive: For any angle A, ™A £™A.
Symmetric: If ™A £ ™B, then ™B £™A.
Transitive: If ™A £ ™B and ™B £™C, then
™A £™C. (p. 109)

2.3 Right Angle Congruence Theorem All right
angles are congruent. (p. 110)

2.4 Congruent Supplements Theorem If two angles
are supplementary to the same angle (or to
congruent angles) then they are congruent. (p. 111)

2.5 Congruent Complements Theorem If two angles
are complementary to the same angle (or to
congruent angles) then the two angles are congruent.
(p. 111)

2.6 Vertical Angles Theorem Vertical angles are
congruent. (p. 112)

3.1 If two lines intersect to form a linear pair of
congruent angles, then the lines are perpendicular.
(p. 137)

3.2 If two sides of two adjacent acute angles are
perpendicular, then the angles are complementary.
(p. 137)

3.3 If two lines are perpendicular, then they intersect to
form four right angles. (p. 137)

3.4 Alternate Interior Angles If two parallel lines are
cut by a transversal, then the pairs of alternate
interior angles are congruent. (p. 143)

3.5 Consecutive Interior Angles If two parallel lines
are cut by a transversal, then the pairs of
consecutive interior angles are supplementary.
(p. 143)

3.6 Alternate Exterior Angles If two parallel lines are
cut by a transversal, then the pairs of alternate
exterior angles are congruent. (p. 143)

3.7 Perpendicular Transversal If a transversal is
perpendicular to one of two parallel lines, then it is
perpendicular to the other. (p. 143)

3.8 Alternate Interior Angles Converse If two lines
are cut by a transversal so that alternate interior
angles are congruent, then the lines are parallel.
(p. 150)

3.9 Consecutive Interior Angles Converse If two
lines are cut by a transversal so that consecutive
interior angles are supplementary, then the lines are
parallel. (p. 150)

3.10 Alternate Exterior Angles Converse If two lines
are cut by a transversal so that alternate exterior
angles are congruent, then the lines are parallel.
(p. 150)

3.11 If two lines are parallel to the same line, then they
are parallel to each other. (p. 157)
3.12 In a plane, if two lines are perpendicular to the same
line, then they are parallel to each other. (p. 157)

4.1 Triangle Sum Theorem The sum of the measures
of the interior angles of a triangle is 180°. (p. 196)
Corollary The acute angles of a right triangle are
complementary. (p. 197)

4.2 Exterior Angle Theorem The measure of an
exterior angle of a triangle is equal to the sum of the
measures of the two nonadjacent interior angles.
(p. 197)

4.3 Third Angles Theorem If two angles of one
triangle are congruent to two angles of another
triangle, then the third angles are also congruent.
(p. 203)

4.4 Reflexive Property of Congruent Triangles
Every triangle is congruent to itself.
Symmetric Property of Congruent Triangles
If ¤ABC £¤DEF, then ¤DEF £¤ABC.
Transitive Property of Congruent Triangles
If ¤ABC £¤DEF and ¤DEF £¤JKL, then
¤ABC £¤JKL. (p. 205)

4.5 Angle-Angle-Side (AAS) Congruence Theorem If
two angles and a nonincluded side of one triangle
are congruent to two angles and the corresponding
nonincluded side of a second triangle, then the two
triangles are congruent. (p. 220)

4.6 Base Angles Theorem If two sides of a triangle are
congruent, then the angles opposite them are
congruent. (p. 236)
Corollary If a triangle is equilateral, then it is
equiangular. (p. 237)

4.7 Converse of the Base Angles Theorem If two
angles of a triangle are congruent, then the sides
opposite them are congruent. (p. 236)
Corollary If a triangle is equiangular, then it is
equilateral. (p. 237)

4.8 Hypotenuse-Leg (HL) Congruence Theorem If
the hypotenuse and a leg of a right triangle are
congruent to the hypotenuse and a leg of a second
right triangle, then the two triangles are congruent.
(p. 238)

5.1 Perpendicular Bisector Theorem If a point is on a
perpendicular bisector of a segment, then it is
equidistant from the endpoints of the segment.
(p. 265)

5.2 Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a
segment, then it is on the perpendicular bisector of
the segment. (p. 265)

5.3 Angle Bisector Theorem If a point is on the
bisector of an angle, then it is equidistant from the
two sides of the angle. (p. 266)

5.4 Converse of the Angle Bisector Theorem If a
point is in the interior of an angle and is equidistant
from the sides of the angle, then it lies on the
bisector of the angle. (p. 266)

5.5 Concurrency of Perpendicular Bisectors of a
Triangle The perpendicular bisectors of a triangle
intersect at a point that is equidistant from the
vertices of the triangle. (p. 273)

5.6 Concurrency of Angle Bisectors of a Triangle
The angle bisectors of a triangle intersect at a point
that is equidistant from the sides of the triangle.
(p. 274)

5.7 Concurrency of Medians of a Triangle The
medians of a triangle intersect at a point that is two
thirds of the distance from each vertex to the
midpoint of the opposite side. (p. 279)

5.8 Concurrency of Altitudes of a Triangle The lines
containing the altitudes of a triangle are concurrent.
(p. 281)

5.9 Midsegment Theorem The segment connecting the
midpoints of two sides of a triangle is parallel to the
third side and is half as long. (p. 288)

5.10 If one side of a triangle is longer than another side,
then the angle opposite the longer side is larger than
the angle opposite the shorter side. (p. 295)

5.11 If one angle of a triangle is larger than another
angle, then the side opposite the larger angle is
longer than the side opposite the smaller angle.
(p. 295)

5.12 Exterior Angle Inequality The measure of an
exterior angle of a triangle is greater than the
measure of either of the two nonadjacent interior
angles. (p. 296)

5.13 Triangle Inequality The sum of the lengths of any
two sides of a triangle is greater than the length of
the third side. (p. 297)

5.14 Hinge Theorem If two sides of one triangle are
congruent to two sides of another triangle, and the
included angle of the first is larger than the included
angle of the second, then the third side of the first is
longer than the third side of the second. (p. 303)

5.15 Converse of the Hinge Theorem If two sides of
one triangle are congruent to two sides of another
triangle, and the third side of the first is longer than
the third side of the second, then the included angle
of the first is larger than the included angle of the
second. (p. 303)

6.1 Interior Angles of a Quadrilateral The sum of th
measures of the interior angles of a quadrilateral is
360°. (p. 324)

6.2 If a quadrilateral is a parallelogram, then its
opposite sides are congruent. (p. 330)

6.3 If a quadrilateral is a parallelogram, then its
opposite angles are congruent. (p. 330)

6.4 If a quadrilateral is a parallelogram, then its
consecutive angles are supplementary. (p. 330)

6.5 If a quadrilateral is a parallelogram, then its
diagonals bisect each other. (p. 330)

6.6 If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
(p. 338)

6.7 If both pairs of opposite angles of a quadrilateral ar
congruent, then the quadrilateral is a parallelogram.
(p. 338)

6.8 If an angle of a quadrilateral is supplementary to
both of its consecutive angles, then the quadrilatera
is a parallelogram. (p. 338)

6.9 If the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram. (p. 338)

6.10 If one pair of opposite sides of a quadrilateral are
congruent and parallel, then the quadrilateral is a
parallelogram. (p. 340)
Rhombus Corollary A quadrilateral is a rhombus
if and only if it has four congruent sides. (p. 348)
Rectangle Corollary A quadrilateral is a rectangle
if and only if it has four right angles. (p. 348)
Square Corollary A quadrilateral is a square if and
only if it is a rhombus and a rectangle. (p. 348)

6.11 A parallelogram is a rhombus if and only if its
diagonals are perpendicular. (p. 349)

6.12 A parallelogram is a rhombus if and only if each
diagonal bisects a pair of opposite angles. (p. 349)

6.13 A parallelogram is a rectangle if and only if its
diagonals are congruent. (p. 349)

6.14 If a trapezoid is isosceles, then each pair of base
angles is congruent. (p. 356)

6.15 If a trapezoid has a pair of congruent base angles,
then it is an isosceles trapezoid. (p. 356)
6.16 A trapezoid is isosceles if and only if its diagonals
are congruent. (p. 356)

6.17 Midsegment Theorem for Trapezoids The
midsegment of a trapezoid is parallel to each base
and its length is one half the sum of the lengths of
the bases. (p. 357)

6.18 If a quadrilateral is a kite, then its diagonals are
perpendicular. (p. 358)

6.19 If a quadrilateral is a kite, then exactly one pair of
opposite angles are congruent. (p. 358)

6.20 Area of a Rectangle The area of a rectangle is the
product of its base and height. A = bh (p. 372)

6.21 Area of a Parallelogram The area of a
parallelogram is the product of a base and its
corresponding height. A = bh (p. 372)

6.22 Area of a Triangle The area of a triangle is one
half the product of a base and its corresponding
height.
A = 1
2
bh (p. 372)

6.23 Area of a Trapezoid The area of a trapezoid is one
half the product of the height and the sum of the
bases.
A = 1
2
h(b1 + b2 ) (p. 374)

6.24 Area of a Kite The area of a kite is one half the
product of the lengths of its diagonals. A = 1
2
d1d2
(p. 374)

6.25 Area of a Rhombus The area of a rhombus is equal
to one half the product of the lengths of the
diagonals. A = 1
2
d1d2 (p. 374)
7.1 Reflection Theorem A reflection is an isometry.
(p. 404)

7.2 Rotation Theorem A rotation is an isometry.
(p. 412)

7.3 If lines k and m intersect at point P, then a reflection
in k followed by a reflection in m is a rotation about
point P. The angle of rotation is 2x°, where x° is the
measure of the acute or right angle formed by k and
m. (p. 414)

7.4 Translation Theorem A translation is an isometry.
(p. 421)

7.5 If lines k and m are parallel, then a reflection in line
k followed by a reflection in line m is a translation.
If Pfl is the image of P, then the following is true:
(1) PPfl
¯ ˘is perpendicular to k and m. (2) PPfl=2d,
where d is the distance between k and m. (p. 421)

7.6 Composition Theorem The composition of two
(or more) isometries is an isometry. (p. 431)

8.1 If two polygons are similar, then the ratio of their
perimeters is equal to the ratios of their
corresponding side lengths. (p. 475)

8.2 Side-Side-Side (SSS) Similarity Theorem If the
lengths of the corresponding sides of two triangles
are proportional, then the triangles are similar.
(p. 488)

8.3 Side-Angle-Side (SAS) Similarity Theorem If an
angle of one triangle is congruent to an angle of a
second triangle and the lengths of the sides
including these angles are proportional, then the
triangles are similar. (p. 488)

8.4 Triangle Proportionality Theorem If a line
parallel to one side of a triangle intersects the other
two sides, then it divides the two sides
proportionally. (p. 498)

8.5 Converse of the Triangle Proportionality
Theorem If a line divides two sides of a triangle
proportionally, then it is parallel to the third side.
(p. 498)

8.6 If three parallel lines intersect two transversals, then
they divide the transversals proportionally. (p. 499)

8.7 If a ray bisects an angle of a triangle, then it divides
the opposite side into segments whose lengths are
proportional to the lengths of the other two sides.
(p. 499)

9.1 If an altitude is drawn to the hypotenuse of a right
triangle, then the two triangles formed are similar to
the original triangle and to each other. (p. 527)

9.2 In a right triangle, the altitude from the right angle
to the hypotenuse divides the hypotenuse into two
segments. The length of the altitude is the geometric
mean of the lengths of the two segments. (p. 529)

9.3 In a right triangle, the altitude from the right angle
to the hypotenuse divides the hypotenuse into two
segments. Each leg of the right triangle is the
geometric mean of the hypotenuse and the segment
of the hypotenuse that is adjacent to the leg.
(p. 529)

9.4 Pythagorean Theorem In a right triangle, the
square of the length of the hypotenuse is equal to the
sum of the squares of the lengths of the legs.
(p. 535)

9.5 Converse of the Pythagorean Theorem If the
square of the length of the longest side of a triangle
is equal to the sum of the squares of the lengths of
the other two sides, then the triangle is a right
triangle. (p. 543)

9.6 If the square of the length of the longest side of a
triangle is less than the sum of the squares of the
lengths of the other two sides, then the triangle is
acute. (p. 544)

9.7 If the square of the length of the longest side of a
triangle is greater than the sum of the squares of the
length of the other two sides, then the triangle is
obtuse. (p. 544)

9.8 45°-45°-90° Triangle Theorem In a 45°-45°-90°
triangle, the hypotenuse is 2  times as long as each
leg. (p. 551)

9.9 30°-60°-90° Triangle Theorem In a 30°-60°-90°
triangle, the hypotenuse is twice as long as the
shorter leg, and the longer leg is 3  times as long
as the shorter leg. (p. 551)

10.1 If a line is tangent to a circle, then it is
perpendicular to the radius drawn to the point of
tangency. (p. 597)

10.2 In a plane, if a line is perpendicular to a radius of a
circle at its endpoint on the circle, then the line is
tangent to the circle. (p. 597)

10.3 If two segments from the same exterior point are
tangent to a circle, then they are congruent. (p. 598)

10.4 In the same circle, or in congruent circles, two
minor arcs are congruent if and only if their
corresponding chords are congruent. (p. 605)

10.5 If a diameter of a circle is perpendicular to a chord,
then the diameter bisects the chord and its arc.
(p. 605)

10.6 If one chord is a perpendicular bisector of another
chord, then the first chord is a diameter. (p. 605)

10.7 In the same circle or in congruent circles, two
chords are congruent if and only if they are
equidistant from the center. (p. 606)

10.8 If an angle is inscribed in a circle, then its measure
is half the measure of its intercepted arc. (p. 613)

10.9 If two inscribed angles of a circle intercept the same
arc, then the angles are congruent. (p. 614)

10.10 If a right triangle is inscribed in a circle, then the
hypotenuse is a diameter of the circle. Conversely, if
one side of an inscribed triangle is a diameter of the
circle, then the triangle is a right triangle and the
angle opposite the diameter is the right angle.
(p. 615)

10.11 A quadrilateral can be inscribed in a circle if and
only if its opposite angles are supplementary.
(p. 615)

10.12 If a tangent and a chord intersect at a point on a
circle, then the measure of each angle formed is one
half the measure of its intercepted arc. (p. 621)

10.13 If two chords intersect in the interior of a circle,
then the measure of each angle is one half the sum
of the measures of the arcs intercepted by the angle
and its vertical angle. (p. 622)

10.14 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 one half the
difference of the measures of the intercepted arcs.
(p. 622)

10.15 If two chords intersect in the interior of a circle,
then the product of the lengths of the segments of
one chord is equal to the product of the lengths of
the segments of the other chord. (p. 629)

10.16 If two secant segments share the same endpoint
outside a circle, then the product of the length of
one secant segment and the length of its external
segment equals the product of the length of the other
secant segment and the length of its external
segment. (p. 630)

10.17 If a secant segment and a tangent segment share an
endpoint outside a circle, then the product of the
length of the secant segment and the length of its
external segment equals the square of the length of
the tangent segment. (p. 630)

About these ads

Entry filed under: Geometry, Personal Notes. Tags: .

Geometry – Postulates Chemistry – Solublity Rules

32 Comments Add your own

  • 1. nikki  |  February 19, 2008 at 4:12 pm

    this site is really useful for geometry homework when i dont have my theorems or book at home

    Reply
  • 2. jose paolo b. samson  |  August 17, 2008 at 1:25 am

    i like this
    this is a big help for my project
    yhank you

    Reply
  • 3. jessa mae  |  August 25, 2008 at 3:12 am

    jst wanna say thank you..
    ds s such a vry bg help..

    Reply
  • 4. yetijh  |  October 5, 2008 at 12:11 am

    well, it HELPED SOO MUCH, i only had to type like 20 of the 100 THANKS TO YOU, MY HAND ISN’T BROKEN

    Reply
  • 5. matthew  |  October 20, 2008 at 8:59 pm

    thank you so much u saved my life i was able to actually do my math hmwk

    Reply
  • 6. jeff  |  October 23, 2008 at 9:44 pm

    what the heck does “™A £ ™B” mean?????

    Reply
    • 7. Britt  |  November 2, 2009 at 8:03 pm

      If you’re talking about 4.4…
      £ means “congruent to”
      and the star thing means “angle”

      ¤DEF £¤ABC.
      angle DEF is congruent to angle ABC

      Reply
  • 8. Savannah  |  November 11, 2008 at 7:16 pm

    OH MY GOSH!!!!!

    thank you SO much!!!

    this helped me a TON!!!!

    i really needed this for my test!!!!

    you are a lifesaver :)

    Reply
  • 9. DUDE  |  December 18, 2008 at 7:47 pm

    DUDE WOW ITZ HELPFUL, YEA BUT KEEP IT @ URSELVES OK
    LIK DUDE
    LIK DUDE

    Reply
  • 10. ilovethis  |  January 11, 2009 at 11:06 pm

    This was so easy thanks alot man thank you

    Reply
  • 11. Tiarra  |  February 19, 2009 at 6:37 pm

    this is a great thing to use and then it helps me on my homework that i do for skool.

    Reply
  • 12. Amy  |  March 17, 2009 at 10:26 pm

    Thank you sooo much!!! This is great!! :D

    Reply
  • 13. Megan  |  May 1, 2009 at 12:25 am

    hehe this wud b useful except by the time i found this we were on chapter 11 which u dont have

    Reply
  • 14. George  |  May 4, 2009 at 8:12 pm

    Yeah… u should complete it

    Reply
  • 15. Nessa  |  May 17, 2009 at 11:00 pm

    this just helped me complete mi study guide for xams…i diDn’t have the book… THANK YOU!!!!

    Reply
  • 16. cmislang  |  May 19, 2009 at 12:18 am

    HAHA this saved my life from writing.
    thank you! :)

    Reply
  • 18. Ashlica  |  September 15, 2009 at 7:29 am

    Hi! Ilike this site. but i think you should make it like a hyphothesis statement…like “if the triangle is cut by a transversal, then it is parallel” in this way, i’m sure it will be understandable. well anyway, it’s a big help. thanks.

    Reply
  • 19. Ashlica  |  September 15, 2009 at 7:33 am

    oh sorry! i didn’t see the other statements. it was in a hypothesis form. but it’s better if you make all of the theorems and postulates in a hypothesis form. the reason as to why i’m suggesting you this one because it’s better to understand. and i’m sure of that. once again, thank you so much.

    Reply
  • 20. LaNdoi  |  October 18, 2009 at 6:09 am

    tnx.for the theorem.

    Reply
  • 21. olga  |  December 29, 2009 at 5:12 pm

    amazing helped so much….. so i coud go out and play in hawaii

    Reply
  • 22. cadesha  |  May 16, 2010 at 6:55 pm

    omg your awsome…this is a life saver…

    Reply
  • 23. vyshak s  |  July 18, 2010 at 5:58 am

    r u a fool to type prepare this…stupid,son of a bitch….raskal

    Reply
  • 24. beybe svethy  |  August 11, 2010 at 7:40 am

    love it thanks
    i have now my projects
    ty:))))

    Reply
  • 25. floramae  |  September 26, 2010 at 7:23 am

    this page helps students ,especially high school students

    Reply
  • 26. mhaell gwapo  |  October 18, 2011 at 5:50 am

    thank you for this page, because it helps for my project in theorems! oh yeah!!!
    MENDELIANZ!
    in Liloy National High School(LNHS)

    Reply
  • 27. Ziad Arafat  |  January 26, 2012 at 5:09 am

    thank you sooooooooooo much i love you i lost all my geometry books this helped soooooooooooo much

    Reply
  • 28. FIGUEIREDO  |  May 5, 2012 at 5:40 am

    OBRIGADO PELA AJUDA

    Reply
  • 29. Sean Imes  |  May 11, 2012 at 8:44 am

    this helped a ton, thank you!!!!!!

    Reply
  • 30. music education  |  July 15, 2013 at 3:26 am

    Hi there Dear, are you really visiting this web site daily, if so
    after that you will without doubt take fastidious knowledge.

    Reply
  • 31. Alonzo  |  August 4, 2013 at 5:09 am

    This design is spectacular! You obviously know how to keep a reader
    amused. Between your wit and your videos, I was almost moved to start my own blog
    (well, almost…HaHa!) Wonderful job. I really enjoyed what you had to say,
    and more than that, how you presented it. Too cool!

    Reply
  • 32. Francisca  |  August 10, 2014 at 8:49 pm

    Hi to every , since I am genuinely eager of reading this weblog’s post to be updated daily.
    It carries fastidious material.

    Reply

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Trackback this post  |  Subscribe to the comments via RSS Feed


Categories


Follow

Get every new post delivered to your Inbox.

%d bloggers like this: