# GEOMETRY DEFINITIONS

**THEOREMS**

**Theorem 1**. When two lines cross,

(a) adjacent angles add up to 180° , and

(b) vertical angles are equal.

**Theorem 2**. Suppose that ℓ and m are two lines crossed by a transversal.

(a) If ℓ and m are parallel, then both pairs of alternate interior angles are
equal. If at

least one pair of alternate interior angles are equal, then ℓ and m are
parallel.

(b) If ℓ and m are parallel, then each pair of interior angles on the same side
of the

transversal adds up to 180° . If at least one pair of interior angles on the same
side

of the transversal adds up to 180° , then ℓ and m are parallel.

(c) If ℓ and m are parallel, then each pair of exterior angles on the same side
of the

transversal adds up to 180° . If at least one pair of exterior angles on the same
side

of the transversal adds up to 180° , then ℓ and m are parallel.

**Theorem 3.** The angles of a triangle add up to 180° .

**Theorem 4.** If two triangles ABC and DEF have
∠A = ∠D and ∠B = ∠E then also

∠C = ∠F.

**Theorem 5.** (a) If two sides of a triangle are equal then the opposite angles are
equal.

(b) If two angles of a triangle are equal then the opposite sides are equal.

**Theorem 6.** In triangle ABC, if ∠B is a right angle then the area of the triangle
is

1/2AB ∙ BC.

**Theorem 7. **The area of a triangle is one-half of the base times the height.

**Theorem 8** (Pythagorean theorem). In a right triangle the sum of the squares of
the

two legs is equal to the square of the hypotenuse .

**Theorem 9 **(Hypotenuse-Leg Theorem). In triangles ABC and DEF, if ∠A and ∠D

are right angles, and if BC = EF and AB = DE, then ΔABC
ΔDEF. That is, if

two right triangles have the hypotenuse and a leg matching then they are
congruent.

**Theorem 10. **If ABCD is a parallelogram then opposite sides of ABCD are equal.

**Theorem 11.** If ABCD is a parallelogram then opposite angles of ABCD are equal.

**Theorem 12.** If a quadrilateral has a pair of sides which are equal and parallel
then it

is a parallelogram.

**Theorem 13.** A quadrilateral is a parallelogram
the diagonals bisect each
other

(that is, the intersection of the two diagonals is the midpoint of each
diagonal).

**Theorem 14.** (a) Suppose that C is a point on the segment AB. C is the midpoint
of

AB AB = 2AC.

(b) A line segment can have only one midpoint.

**Theorem 15.** In triangle ABC, let D be the midpoint of AC and suppose that E is a

point on BC with DE parallel to AB. Then E is the midpoint of BC and DE = 1/2AB.

**Theorem 16.** In triangle ABC, let D be the midpoint of AC and let E be the
midpoint

of BC. Then DE is parallel to AB and DE = (1/2)AB.

**Theorem 17. **[SAS for similarity] In triangles ABC and DEF, if ∠C = ∠F and

then ΔABC ~ ΔDEF.

**Theorem 18 **(SSS for similarity). In triangles ABC and DEF, if

then ΔABC ~ ΔDEF.

**Theorem 19.** sin(∠ABC) = sin(180°
− ∠ABC)

**Theorem 20.** For any triangle ABC, the area can be calculated by any of the
following

three formulas :

area of ΔABC = 1/2AB ∙ AC sin∠A

area of ΔABC = 1/2AB ∙ BC sin∠B

area of ΔABC = 1/2AC ∙ BC sin∠C

that is, the area is one half the product of two sides times the sine of the included angle.

T**heorem 21** (Law of Sines). In any triangle ABC,

**Theorem 22**. For any triangle ABC, the perpendicular bisectors of AB, AC and BC

are concurrent.

**Theorem 23.** For any triangle ABC, the bisectors of ∠A, ∠B and ∠C are concurrent.

**Theorem 24. **For any triangle ABC, the three altitudes are concurrent.

**Theorem 25.** The point where two medians of a triangle intersect is 2/3 of the way

from each of the two vertices to the opposite midpoint.

**Theorem 26.** For any triangle ABC, the three medians are concurrent.

**Theorem 27.** Let ABC be any triangle. Let O be the circumcenter of ABC, let G be

the centroid of ABC, and let H be the orthocenter of ABC. Then O, G and H are

collinear.

**Theorem 28** (Theorem of Menelaus). Let ABC be any triangle. Let A' be a point of

other than B and C, let B' be a point of
other than A and C, and let C' be a

point of other than A and B. If A', B' and C' are collinear then

**Theorem 29.** The number
is positive if C ' is outside of the segment AB and

negative if C ' is inside AB.

**Theorem 30.** The number
is equal to
if C' is outside of the segment AB and

is equal to
if C' is inside the segment AB.

**Theorem 31.** Let ℓ be any line, and let C', C", A and B be points on ℓ, with A
not the

same point as B. If
then C' is the same point as C".

**Theorem 32.** Let ABC be a triangle. Let A' be a point of
other than B and C, let

B' be a point of other than A and C, and let C' be a point of
other than A and

B. If A', B' and C' are collinear then

**Theorem 33.** Let ABC be a triangle. Let A' be a point of
other than B and C, let

B' be a point of other than A and C, and let C' be a point of
other than A and

B. If

then A', B' and C' are collinear.

**Theorem 34** (Theorem of Ceva). Let ABC be a triangle. Let A' be a point of
other

than B and C, let B' be a point of other than A and C, and let C' be a point of

other than A and B. If are concurrent then

**Theorem 35.** Let ABC be a triangle. Let A' be a point of
other than B and C, let

B' be a point of other than A and C, and let C' be a point of
other than A and

B. If

then are concurrent.

**Theorem 36.** Let A, B and C be points on a circle with center O.

(a) If B is outside of ∠AOC, then ∠ABC =(1/2)∠AOC.

(b) If B is inside of ∠AOC, then ∠ABC = 180° − (1/2)∠AOC.

**Theorem 37.** The arc cut off by a central angle is the same number of degrees as
the

angle.

**Theorem 38. **Let A, B and C be points on a circle, and let ADC be the arc cut o
by

∠ABC. Then

that is, the number of degrees in angle ABC is half the number of degrees in arc ADC.

**Theorem 39.** Let A, B, C and D be points on a circle and consider the angles ABC

and ADC.

(a) If ∠ABC and ∠ADC cut off the same arc, then ∠ABC = ∠ADC.

(b) If ∠ABC and ∠ADC do not cut o the same arc, then ∠ABC = 180° − ∠ADC.

**Theorem 40. **Let A, B, C and D be points on a circle, and suppose that the lines
AB

and CD meet at a point P. Then PA ∙ PB = PC ∙ PD.

**Theorem 41.** Let C be a circle with center O, let A be a point on the circle, and
let m

be a line through A. Then m is tangent to the circle
m is perpendicular to
OA.

**Theorem 42.** Let A and B be points on a circle, let C be a point on the tangent
line at

B, and let ADB be the arc cut off by ∠ABC. Then

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