GEOMETRY THEOREMS Flashcards
Parts Whole Theorem for Segments
1) If AB = CD and AE = CF, then FD = EB
2) If AE = CF and FD = EB, then AB = CD
C_____F________D
A_____E________B
Parts Whole Theorem for Angles
2 on the theorem list - same as parts whole for segments
Midpoint Theorem
If M is the midpoint of AB them AM = 1/2AB
Angle Bisector Theorem
If ray BD bisects angle ABC then angle ABD = 1/2 angle ABC
Halves Whole Theorem for Segments
Let M be the midpoint of AB and N be the midpoint of CD; 1) If AB=CD, them AM = CM 2) If AM = CN, then MB=ND and AB = CD A\_\_\_\_\_\_\_\_\_\_M\_\_\_\_\_\_\_\_\_\_B C\_\_\_\_\_\_\_\_\_\_N\_\_\_\_\_\_\_\_\_\_D
Halves Whole Theorem for Angles
6 on the theorem list - same as halves whole theorem for segments
Common Segment Theorem
1) If AB = CD then AC = BD
2) If AC = BD then AB = CD
A_____B__________C_____D
Common Angle Theorem
8 on the theorem list - same as common segment theorem
Vertical Angle Theorem (VAT)
Verticle Angles are Congruent
Perpendicular Line Theorem
If any one of the following statements about two intersecting lines m and n is true, then all the statements are true
-
1 - 2
----------
4 - 3
-
1) m is perpendicular to n
2) angle 1 = angle 2 (adjacent angles are congruent)
3) angle 1 is a right angle (any angle is right)
4) angle 1 is 90 degrees ( any angle has 90 degrees
Congruent Complements Theorem
If two angles are complements of congruent angles (or the same angle), then those angles are congruent
Congruent Supplements Theorem
If two angles are supplements of congruent angles ( or the same angle), then those two angles are congruent
Parallel Lines Imply Corresponding Angles congruent postulate
Abbreviation: CAPP or // –> corr angles congruent
If two parallel lines are cut by a transversal, then corresponding angles are congruent
Parallel Lines Imply Alternate Interior Angles Congruent
Abbreviation: // –> alt int angles congruent
If two parallel lines are cut by a transversal, then alternate interior angles are congruent
Parallel Lines Imply Alternate Exterior Angles Congruent
Abbreviation: // –> alt ext angles congruent
If two parallel lines are cut by a transversal, then alternate exterior angles are congruent
Parallel Lines Imply Same Side Interior Angles Supplementary
Abbreviation: Same Side Int Sup
If two parallel lines are cut by a transversal, then same side interior angles are supplementary
Parallel Lines Imply Alternate Same Side Exterior Angles Supplementary
Abbreviation: // –> same side ext. sup
If two parallel lines are cut by a transversal, them same side exterior angles are supplementary
Corresponding Angles are Congruent Implies Lines Parallel Postulate
Abbreviation: CCAP or corr angles are congruent –> //
If two lines are cut by a transversal and corresponding angles are congruent then the lines are parallel.
Alternate Interior Angles (congruent) Implies Lines Parallel
If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel
Same Side Interior Angles Supplementary Implies Lines Parallel
Abbreviation: Same Side Sup –> //
If two lines are cut by a transversal and same side interior angles are supplementary, then the lines are parallel
If two lines are perpendicular to the same line
they are parallel
Transitivity with parallel lines:
If line 1// line 2 and line 2 // line 3 then line 1 // line 3
Triangle Sum Theorem
The sum of the measures of the angles of a triangle is 180 degrees
Corollaries to the Triangle Sum Theorem:
1) Remaining Angle in a Triangle Theorem
If two angles of one triangle are congruent to two angles of another congruent to two angles of another triangle, then the third angles are congruent
Corollaries to the Triangle Sum Theorem
2) Each angle of an equiangular triangle has measure 60°.
Corollaries to the Triangle Sum Theorem
3) In a triangle, there can be at most one right or obtuse angle.
Corollaries to the Triangle Sum Theorem
4) The acute angles of a right triangle are complementary.
Exterior Angle of a Triangle Theorem
The measure of an exterior angle of a triangle
equals the sum of the measures of the two remote interior angles.
Polygon Angle Sum Theorem
The sum of the measures of the interior angles of a
convex (or concave!) polygon with n sides is: (n − 2)180.
Polygon Exterior Angle Sum Theorem
The sum of the measures of the exterior
angles of any convex polygon, one angle at each vertex, is 360.
Definition of and ideas about congruent triangles from pages 117-8
Corresponding Parts of Congruent Triangles are Congruent (i.e. Corr. parts of ≅ Δ’s are ≅ )
SSS Postulate
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent
SAS Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent
ASA Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Isosceles Triangle Theorem
Two sides of a triangle are congruent if and only if the
angles opposite those sides (the base angles) are congruent.
Corollaries of the Isosceles Triangle Theorem
1) A triangle is equilateral if and only if it is equiangular.
2) An equilateral triangle has three 60° angles.
3) The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.
AAS
If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
HL
If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.
Magic Line Theorem or Isosceles-Median Theorem
If any two of the following statements about ΔABC with point M on BC are true, then all four statements are true 1) AB = AC 2) Angle BAM = Angle CAM 3) MB = MC 4) AM is perpendicular to BC (Diagram on Theorem List - #35)
Parallelogram ⇒ opp. sides ≅
The opposite sides of a parallelogram are congruent.
Parallelogram ⇒ opp. ∠’s ≅
The opposite angles of a parallelogram are congruent.
Parallelogram ⇒ diagonals bisect each other
The diagonals of a parallelogram bisect each other.
Parallelogram ⇒ consec∠’s supplementary
Consecutive angles in a parallelogram are supplementary.
Opp. sides quad. ≅ ⇒ Parallelogram
If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Opp.∠’s ≅ ⇒ Parallelogram
If the opposite angles of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
Opp. sides quad. ≅ and ⇒ Parallelogram
If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.
Diagonals bisect each other ⇒ Parallelogram
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Consec∠’s supp. ⇒ Parallelogram
If consecutive angles of a quadrilateral are supplementary, then the quadrilateral is a parallelogram.
Midsegment Theorem
The segment that joins the midpoints of two sides of a
triangle is parallel to the third side and is half as long as the third side (Theorem List #45)
Corollary to ≅ Cutter Theorem
A line that contains the midpoint of one side of a
triangle and is parallel to another side passes through the midpoint of the third side
Inner Alien or (Midpoint Quadrilateral Theorem)
The quadrilateral formed by joining the midpoints of the sides of any quadrilateral is a parallelogram.
rectangle → ≅ diagonals
The diagonals of a rectangle are congruent.
parallelogram + ≅ diagonals → rectangle
If a parallelogram has congruent diagonals, then it is a rectangle.
parallelogram + right ∠ → rectangle
If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
2 consec. sides ≅ + parallelogram → rhombus
If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.
rhombus → ⊥ diagonals
The diagonals of a rhombus are perpendicular.
parallelogram +⊥ diagonals → rhombus
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
rhombus → diagonals bisect each angle
Each diagonal of a rhombus bisects two angles of the rhombus.
diagonals of a parallelogram bisect 1∠→ rhombus
If a diagonal bisects at least one angle of a parallelogram, then the parallelogram is a rhombus.
Isosceles Trapezoid Theorem
1) If a trapezoid is isosceles, then its base angles are congruent.
2) If the base angles of a trapezoid are congruent, then the trapezoid is isosceles.
The diagonals of a trapezoid are congruent if and only if
the trapezoid is isosceles
Trapezoid Median Theorem
The median of a trapezoid is parallel to both bases and
has a length equal to the average of the base lengths. (i.e in Trapezoid ABCD with median MN, AB // MN // CD and MN = 1/2 (AB + CD)
Whole > Part Theorem
If a > 0, b > 0, and a + b = c, then c > a and c > b.
Mo mo le le
If a > b and c > d then a + c > b + d
Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure or either remote interior angle.
Alligator Theorem (longer side ó bigger opp. angle)
If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle
opposite the second side.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
AA ~ Post
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
3 in 1 Right Triangles
If the 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.
The Pythagorean Theorem
In a right triangle, the square of the hypotenuse is equal
to the sum of the squares of the legs.
Converse of the Pythagorean Theorem
If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
45-45-90 Triangle Theorem
the hypotenuse is 2 times as long as a leg
30-60-90 Triangle Theorem
the hypotenuse is twice as long as the shorter leg, and
the longer leg is 3 times as long as the shorter leg
Sine, Cosine, and Tangent Ratios
Let x be an acute (not the right) angle in a right triangle. Then, sin(x) = opposite side/hypotenuse cos(x) = adjacent side/hypotenuse tan(x) = opposite side/adjacent side Remember: Soh Cah Toa
Tangent ↔ ⊥ to radius (Lauren’s Theorem)
A line is tangent to a circle, if and only if the line is perpendicular to the radius drawn to the point of tangency.
≅ tangents (Watermelon Graduation Cap)
Tangents from the same point to a circle are congruent.
Arc add. post.
The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.
arcs are ≅ ⟺ central ∠s are ≅ (Defn of ≅ arcs)
In the same circle or in congruent circles, two arcs are congruent if and only if their central angles are congruent.
Arcs ≅ ⟺ chords ≅
In the same circle or in congruent circles, two arcs are
congruent if and only if they have congruent chords.
⊥ diameter bisects the chord and arc
A diameter that is perpendicular to a chord bisects the chord and its arc.
diameter bisects the chord and arc ⇒ ⊥
if a diameter bisects the chord and arc then it is perpendicular to the chord
In the same circle or congruent circles, chords equidistant from the center (or centers) are ≅.
Also, ≅ chords are equidistant from the center (or centers).
Inscribed Angle Theorem
The measure of an inscribed angle is equal to half the measure of its intercepted arc.
Corollaries of the Inscribed Angle Theorem
1) If two inscribed angles intercept the same arc, then the angles are congruent.
2) An angle inscribed in a semicircle is a right angle.
3) If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
Extension of Inscribed Angle Theorem
The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.
Area of a Square
s⋅s (s squared)
Area congruence postulate
If two figures are congruent, then they have the same
area.
Area Addition postulate
The area of a region is the sum of the areas of its nonoverlapping parts.
Area of a rectangle
l ⋅w
Area of a parallelogram
b⋅ h where h is the height of the parallelogram
Area of a Triangle
0.5 (b⋅ h)
Area of a Rhombus
A = 0.5 (d1 ⋅ d2) where d1 and d2 are the diagonals of the rhombus
Area of a Trapezoid
A = 0.5h⋅(b1 + b2 ) or A = h ⋅median
Area of a Circle
A = πr(squared)
Circumference of a Circle
C = 2πr or C = πd
Area of a Sector
= x/360⋅(πrsquared) where x is the arc measure of the arc that forms the sector.
Length of a Sector
= x/360⋅(2πr) where x is the arc measure of the arc that forms the sector.
The distance, d, between the two points x1, y1 and x2, y2 can be found using the following distance formula:
d = radical (x2 - x1)squared + (y2 - y1) squared
The slope of a line between the points x1, y1 and x2, y2
rise over run
delta y over delta x
y2 - y1 over x2 - x1
The slope of a horizontal line is zero, and the slope of a vertical line is undefined.
The Equation of a Circle
(x-a)squared + (y-b) squared = r squared
Center: (a, b)
Radius: r
Parallel lines
:) have equal slopes.
Perpendicular lines have slopes that are
negative reciprocals of one another
The midpoint between the points is the average of the x-coordinates and y coordinates. Given (x1, y1) and (x2, y2) the midpoint is
(x1, y1)/2 + (x2, y2)/2
(x1, y1) divided by two plus (x2, y2) divided two
