Proofs and Theorems

2024 Geometry proofs and theorems to know

Theorem

a mathmatical statement that can be proved

Must be in Two-Column Proof!

1) Given Info.

2) Focus on the Prove Statement

3) Use and look at the diagram

4) Statements

5) Reasoning

2 Angles Rt. Congruent Theorem

If two angles are right angles, then they are congruent

2 Angles Str. Congruent Theorem

If two angles are straight angles, then they are congruent

Supplementary to Same Congruency

If angles are supplementary to the same angle, then they are congruent

Supplementary to Congruent Angles

If angles are supplementary to congruent angles, then they are congruent

Complementary to Same Congruency

If angles are complementary to the same angle, then they are congruent

Complementary to Congruent Angles

If angles are complementary to congruent angles, then they are congruent

Addition Property (4)

• If congruent segments are added to congruent segments, then the sums are congruent

• If congruent angles are added to congruent anlges then the sums are congruent

• If an angle is added to congruent angles, then the sums are congruent.

• If a segment is added to two congruent segments, them the sums are congruent

Subtraction Property (2)

• If a segments (or angles) is subtracted from congruent segments (or angles), then the difference are congruent.

• If congruent segments (or angles) are subtracted from congruent segments (or angles), then the difference are congruent.

Multiplication Property

If segments (or angles) are congruent, then their like multiples are congruent. (Small to big)

Division Property

If segments (or angles) are congruent, then their like divisions are congruent. (Big to small)

Cue words for Mult. and Div. property

Midpoint, bisects, angle bisector/trisector, etc.

VAT

Vertical Angle Theorem

CPCTC

Corrosponding Parts of Congruent Triangles are Congruent

Radii Congruency

All Radii in a circle are congruent

Median (THINK MIDPT.)

A line (or seg) that extends from the vertex of a triangle to the midpt. of the opp. side

Altitude (THINK PERPENDICULAR)

A line (or seg) that extends from the vertex of a triangle and is perpendicular to the opp. side

Auxiliary Lines Postualte

Two pts. determine a line

Concurrent

Lines that meet at exactly one pt. are conncurrent

Orthocenter

Centroid

ITT

(Isosceles Triangle Thereom) Is a theorem stating that if two sides of a triangle are congruent, then the sides opposite those angles are also congruent.

ITTC

(Isosceles Triangle Theorem Converse) If two angles of a triangle are congruent, then the sides opposite those angles are also congruent.

Triangle Congruency YES

• SSS postulate

• SAS postulate

• ASA postulate

• AAS postulate

Triangle Congruency NO

• ASS postulate

• AAA Postulate

Inverse of ITT

If two sides of a triangle are not congruent, then the angles opposite those sides are also not congruent, and the larger angle opposite the longer side.

Inverse ITTC

If two angles of a triangle are not congruent, then the sides opposite those angles are also congruent and the longer side is opposite the larger angle.

HL Postulate

hypotenuseIf there exists a correspondence between the vertices of two right triangles such that the hypotneuse and a leg of one trinalge are congruent to the corresponding parts fo the other triangle, the two right tringles are congruent

RAT

Right Angle Theorem - If 2 angles are both supplementary and congruent, then they are right anlges

Any pt. on perp. bis Theorem

If a pt. is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment

EDT

Equidistance Theorem - If two points are each equidistant from the endpoints of a segment, the the points determine the perpendicular bisector of the segment.

Perpendicular Bisector Postulate

A Perpendicular Bisector is the shortest path between two points.

Method for Using Indirect Proofs

1. Either the prove statement is true or false

2. Assume conclusion is false

3. Continue w/ proof until you get to a contradiction

4. State desired conclusion

When you see negations then —>

Indirect Proofs!

Triangle Interior Angle Theorem

The sum of the measures of the angles in a triangle is 180

Remote Interior Angle Equal Theorem

The measure of an exterior angle equals the sum of the measures of the two remote interior angles

Midline Theorem

A segment joining the midpoints of a triangle is parallel to the third side, and its length is one-half the length of the third side.

No Choice-Theorem

If 2 angles of 1 triangle are congruent to 2 angles of a 2nd triangle, then the 3rd angles are congruent

AAS

If there exists a correspondence between the vertices of 2 triangles such that 2 angles and the NON-INCLUDED side of 1 are congruent to the corresponding parts of the other, then the triangles are congruent.

The sum of a polygon's interior angles with n sides is given by the formula.

180(n-2)

The sum of the exterior angles of a polygon, taking one at each vertex, is always ___

360°

The number of diagonals d in an n-sided polygon is given by the formula.

n(n-3) / 2

Each exterior angle E of an equiangular n-sided polygon is given by the formula.

360/n

Individual int angles of regular polygon equation

180(n-2)/n OR 360/n, then answer of that minus 180. Ex. 360/5=72. 180-72=108, aka measure of individual int angles.

MEPT

Means Extremes Product Theorem: a/b = c/d —> a*d = c*b

MERT

Means Extremes Ratio Theorem: a*d = c*b —> a/b = c/d or a/c = b/d (or any of their reciprocals)

Ratio Perimeter 2 Similar Polygons

The ratio of the perimeters of two similar polygons is = to the ratio of any pair of corresponding sides.

AA

Angle Angle - If there exists a correspondence between the vertices of 2 triangles such that two of the angles of one triangle are congruent to the corresponding angles of the other, then the triangles are similar.

SSS~

If there exists a correspondence between the vertices of 2 triangles such that the ratio of the measure of corresponding sides are equal

SAS~

If there exists a correspondence between the vertices of 2 triangles such that the ratio of the measure of corresponding sides are equal, then the triangles are similar.

Side Splitter Theroem

If a line is parallel to 1 side of a triangle and intersects the other two sides, it divides those 2 sides proportionaly. (Look on 8.5 notes for example)

3 parallel lines 2 transversal Theorem

If 3 or more parallel lines are intersected by two transversal, the parallel lines divide the transversals proportionally. (Look on 8.5 notes for example)

(Most imp. Ch. 8 Theorem) Angle Bisector Theorem

If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides [a/b=d/c OR a/d=b/c] (Look on 8.5 notes for example)

Circumference Equation

2πr or dπ

Area Equation

A = πr ²

If an altitude is drawn to the hypotenuse of a right triangle then:

a. All 3 triangles are similar to each other

b. There exists a mean proportional between 2 other sides of the triangles for each of the 3 sets of triangles (Altitude)

c. Either leg of the given triangle set is the mean proportional between the hypotenuse and the given triangle and the projection of that leg

Write the mean proportional (1st one)

Paul Always Ate Oranges:

p/a = a/o

Write the mean proportional (2nd one)

Pretty Little Liars Hate

p/l = l/h

Pythagorean Theorem

a² + b² = c²

Converse Pythagorean Theorem

If a² + b² = c² then the triangle is a right triangle

Right:

Acute:

Obtuse:

a² + b² = c²

a² + b² > c²

a² + b² < c²

Common (Ms. Kiesselbach may use) Triples

(3,4,5) (3 root 3, 5 root 3, 4 root 3) (5,12,13) (7,24,25) (8,15,17) (9,40,41)

Length Equation

degree of arc/360 (2 pie r)

Measure Equation

degree of arc

Formula for length of the diagonals in a rectangular prism

Radius perp. to chord —>

If a radius is perp. to a chord, then it bisects the chord

If radius bis. chord

If radius bis. chord, then it’s perp. to the chord

Perp. bis. of a chord

Perp. bis. of a chord passes through the center of a circle

2 Chords Equidistant Center Circ. then _____

If 2 chords are equidistant from the center of a circle then they are congruent

2 Chords of a circle are congruent then _____

If 2 chords of a circle are congruent then they are equidistant from the center

If arcs of a circle (or congruent circles) are congruent then…

If the arcs of a circle (or congruent circles) are congruent then, their corresponding chords are congruent

If 2 corresponding chords of a circle (or congruent circles) are congruent then…

If corresponding chords of a circle (or congruent circles) are congruent then, their arcs are congruent

If 2 central angles of a circle (or 2 congruent circles) are congruent then…

If 2 central angles of a circle (or 2 congruent circles) are congruent then, their corros. arcs are congruent

If 2 corros. arcs of a circle (or 2 congruent circles) are congruent then…

If 2 corros. arcs of a circle (or 2 congruent circles) are congruent then, their central angles are congruent

If 2 chords in a circle (or 2 congruent circles) are congruent then…

If 2 chords in a circle (or 2 congruent circles) are congruent then, the corros. central angles are congruent

If the corros. central angles in a circle (or 2 congruent circles) are congruent then…

If the corros. central angles in a circle (or 2 congruent circles) are congruent then, the chords are congruent