Proofs and Theorems

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2024 Geometry proofs and theorems to know

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67 Terms

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Theorem

a mathmatical statement that can be proved

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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

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2 Angles Rt. Congruent Theorem

If two angles are right angles, then they are congruent

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2 Angles Str. Congruent Theorem

If two angles are straight angles, then they are congruent

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Supplementary to Same Congruency

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

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Supplementary to Congruent Angles

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

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Complementary to Same Congruency

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

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Complementary to Congruent Angles

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

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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

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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.

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Multiplication Property

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

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Division Property

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

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Cue words for Mult. and Div. property

Midpoint, bisects, angle bisector/trisector, etc.

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VAT

Vertical Angle Theorem

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CPCTC

Corrosponding Parts of Congruent Triangles are Congruent

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Radii Congruency

All Radii in a circle are congruent

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Median (THINK MIDPT.)

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

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Altitude (THINK PERPENDICULAR)

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

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Auxiliary Lines Postualte

Two pts. determine a line

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Concurrent

Lines that meet at exactly one pt. are conncurrent

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<p>Orthocenter</p>

Orthocenter

Centroid

<p>Centroid</p>
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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.

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ITTC

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

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Triangle Congruency YES

  • SSS postulate

  • SAS postulate

  • ASA postulate

  • AAS postulate

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Triangle Congruency NO

  • ASS postulate

  • AAA Postulate

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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.

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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.

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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

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RAT

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

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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

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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.

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Perpendicular Bisector Postulate

A Perpendicular Bisector is the shortest path between two points.

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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

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When you see negations then —>

Indirect Proofs!

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Triangle Interior Angle Theorem

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

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Remote Interior Angle Equal Theorem

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

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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.

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No Choice-Theorem

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

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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.

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

180(n-2)

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

360°

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

n(n-3) / 2

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

360/n

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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.

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MEPT

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

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MERT

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

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Ratio Perimeter 2 Similar Polygons

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

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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.

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SSS~

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

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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.

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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)

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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)

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(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)

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Circumference Equation

2πr or

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Area Equation

A = πr 2

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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

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<p>Write the mean proportional (1st one)</p>

Write the mean proportional (1st one)

Paul Always Ate Oranges:

p/a = a/o

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<p>Write the mean proportional (2nd one)</p>

Write the mean proportional (2nd one)

Pretty Little Liars Hate

p/l = l/h

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Pythagorean Theorem

a2 + b2 = c2

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Converse Pythagorean Theorem

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

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Right:

Acute:

Obtuse:

a2 + b2 = c2

a2 + b2 > c2

a2 + b2 < c2

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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)

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knowt flashcard image
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Length Equation

degree of arc/360 (2 pie r)

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Measure Equation

degree of arc

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Formula for length of the diagonals in a rectangular prism

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