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SUPPLEMENTARY ANGLES. 2-angles that add up to 180 degrees. COMPLEMENTARY ANGLES. 2-angles that add up to 90 degrees. Vertical Angles . Vertical Angles are Congruent to each other.
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SUPPLEMENTARY ANGLES
2-angles that add up to 180 degrees.
COMPLEMENTARY ANGLES
2-angles that add up to 90 degrees
Vertical Angles
Vertical Angles are Congruent to each other
• <1 =<3• <2=<4• <1+<2=180 degrees• <2+<3=180 degrees• <3+<4=180 degrees• <4+<1=180 degrees
PARALLEL LINES CUT BY A TRANSVERSAL
SUM OF THE INTERIOR ANGLES OF A TRIANGLE
180 DEGREES
EQUILATERAL TRIANGLE
Triangle with equal angles and equal sides
ISOSCELESTRIANGLE
TRIANGLE WITH 2 SIDES = AND 2 BASE ANGLES =
ISOSCELES RIGHT TRIANGLE
RIGHT TRIANGLE WITH BC=CA and < A = <B
EXTERIOR ANGLE THEOREM
LARGEST ANGLE OF A TRIANGLE
ACROSS FROM THE LONGEST SIDE
SMALLEST ANGLE OF A TRIANGLE
ACROSS FROM THE SHORTEST SIDE
LONGEST SIDE OF A TRIANGLE
ACROSS FROM THE LARGEST ANGLE
SMALLEST SIDE OF A TRIANGLE
ACROSS FROM THE SMALLEST ANGLE
TRIANGLE INEQUALITY THEOREM
The sum of 2-sides of a triangles must be larger than the 3rd side.
PROPORTIONS IN THE RIGHT TRIANGLE
ya
ac
xbyh
hx
or bc
(Upside down T!!!!)
(Big Angle Small Angle!!!!!)
CONCURRENCY OF THE THE ANGLE BISECTORS
INCENTER
CONCURRENCY OF THE PERPENDICULAR
BISECTORS
CIRCUMCENTER
CONCURRENCY OF THE MEDIANS
CENTROID
MEDIANS ARE IN A RATIO OF 2:1
CONCURRENCY OF THE ALTITUDES
ORTHOCENTER
Properties of a Parallelogram
Parallelogram
• Opposite sides are congruent.• Opposite sides are parallel.• Opposite angles are congruent.• Diagonals bisect each other.• Consecutive (adjacent) angles are
supplementary (+ 180 degrees).• Sum of the interior angles is 360 degrees.
Properties of a Rectangle
Rectangle
• All properties of a parallelogram.• All angles are 90 degrees.• Diagonals are congruent.
Properties of a Rhombus
Rhombus
• All properties of a parallelogram.• Diagonals are perpendicular (form right
angles).• Diagonals bisect the angles.
Properties of a Square
Square
• All properties of a parallelogram.• All properties of a rectangle.• All properties of a rhombus.
Properties of an Isosceles Trapezoid
Isosceles Trapezoid
• Diagonals are congruent.• Opposite angles are supplementary + 180
degrees.• Legs are congruent
Median of a Trapezoid
DISTANCE FORMULA
MIDPOINT FORMULA
SLOPE FORMULA
PROVE PARALLEL LINES
EQUAL SLOPES
PROVE PERPENDICULAR LINES
OPPOSITE RECIPROCAL SLOPES (FLIP/CHANGE)
PROVE A PARALLELOGRAM
Prove a Parallelogram
• Distance formula 4 times to show opposite sides congruent.
• Slope 4 times to show opposite sides parallel (equal slopes)
• Midpoint 2 times of the diagonals to show that they share the same midpoint which means that the diagonals bisect each other.
How to prove a Rectangle
Prove a Rectangle
• Prove the rectangle a parallelogram.• Slope 4 times, showing opposite sides are
parallel and consecutive (adjacent) sides have opposite reciprocal slopes thus, are perpendicular to each other forming right angles.
How to prove a Square
Prove a Square
• Prove the square a parallelogram.• Slope formula 4 times and distance
formula 2 times of consecutive sides.
Prove a Trapezoid
Prove a Trapezoid
• Slope 4 times showing bases are parallel (same slope) and legs are not parallel.
Prove an Isosceles Trapezoid
Prove an Isosceles Trapezoid
• Slope 4 times showing bases are parallel (same slopes) and legs are not parallel.
• Distance 2 times showing legs have the same length.
Prove Isosceles Right Triangle
Prove Isosceles Right Triangle
• Slope 2 times showing opposite reciprocal slopes (perpendicular lines that form right angles) and Distance 2 times showing legs are congruent.
• Or Distance 3 times and plugging them into the Pythagorean Theorem
Prove an Isosceles Triangle
Prove an Isosceles Triangle
• Distance 2 times to show legs are congruent.
Prove a Right Triangle
Prove a Right Triangle
• Slope 2 times to show opposite reciprocal slopes (perpendicular lines form right angles).
Sum of the Interior Angles
180(n-2)
Measure of one Interior Angle
Measure of one interior angle
180( 2)nn
Sum of an Exterior Angle
360 Degrees
Measure of one Exterior Angle
360/n
Number of Diagonals
2)3( nn
2)3( nn
1-Interior < + 1-Exterior < =
180 Degrees
Number of Sides of a Polygon
Ext360
Ext1360
Converse of PQ
Change OrderQP
Inverse of PQ
Negate
~P~Q
Contrapositive of PQ
Change Order and Negate
~Q~PLogically Equivalent: Same
Truth Value as PQ
Negation of P
Changes the truth value
~P
Conjunction
And (^)
P^QBoth are true to be true
Disjunction
Or (V)
P V Qtrue when at least one is true
Conditional
If P then QPQ
Only false when P is true and Q is false
Biconditional
(iff: if and only if)TT =TrueF F = True
Locus from 2 points
The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment
determined by the two points.
Locus of a Line
Set of Parallel Lines equidistant on each side of the line
Locus of 2 Parallel Lines
3rd Parallel Line Midway in between
Locus from 1-Point
Circle
Locus of the Sides of an Angle
Angle Bisector
Locus from 2 Intersecting Lines
2-intersecting lines that bisect the angles that are formed by the intersecting lines
Reflection through the x-axis
(x, y) (x, -y)
Reflection in the y-axis
(x, y) (-x, y)
Reflection in line y=x
(x, y) (y, x)
REFLECTION IN Y=-X
(X, Y) (-Y, -X)
Reflection in the origin
(x, y) (-x, -y)
Rotation of 90 degrees
(x, y) (-y, x)
Rotation of 180 degrees
(x, y) (-x, -y)Same as a reflection in the
origin
Rotation of 270 degrees
(x, y) (y, -x)
Translation of (x, y)
Ta,b(x, y) (a+x, b+y)
Dilation of (x, y)
Dk (x, y) (kx, ky)
Isometry
Isometry: Transformation that Preserves Distance
• Dilation is NOT an Isometry• Direct Isometries • Indirect Isometries
Direct Isometry
Direct Isometry
• Preserves Distance and Orientation (the way the vertices are read stays the same)
• Translation• Rotation
Opposite Isometry
Opposite Isometry
• Distance is preserved• Orientation changes (the way the vertices
are read changes)• Reflection• Glide Reflection
What Transformation is NOT an Isometry?
Dilation
GLIDE REFLECTION
COMPOSITION OF A REFLECTION AND A
TRANSLATION
Area of a Triangle
bh21 Triangle a of Area
bh21 Triangle a of Area
Area of a Parallelogram
Area of a Rectangle
Area of a Trapezoid
)(21 Trapezoid a of Area 21 bbh
)(21 Area 21 bbhTrapezoid
Area of a Circle
Circumference of a Circle
Surface Area of a Rectangular Prism
Surface Area of a Triangular Prism
)()()()21(2 332211 hbhbhbbhSA
Surface Area of a Trapezoidal Prism
)()()()()](21[2 4433221121 hbhbhbhbbbhSA
H
Surface Area of a Cylinder
Surface Area of a Cube
)(6 2SSA
Volume of a Rectangular Prism
Volume of a Triangular Prism
HbhV )21(
Volume of a Trapezoidal Prism
prism theofHeight H trapezoid theofheight h
)](21[ 21
HbbhV
H
Volume of a Cylinder
Volume of a Triangular Pyramid
pyramid theofheight H triangle theofheight h
]21[
31
HbhV
Volume of a Square Pyramid
pyramid theofheight Hsquare a of sideS
][31 2
HSV
Volume of a Cube
cube a of sideS
3
SV
PERIMETER
ADD UP ALL THE SIDES
VOLUME OF A CONE
Hr *31V
CONE) THE OF (HEIGHT H * BASE) THE OFAREA (31V
2
LATERAL AREA OF A CONE
l *r * LA
222 lrh
SURFACE AREA OF A CONE
rlrSA 2
VOLUME OF A SPHERE
SURFACE AREA OF A SPHERE
SIMILAR TRIANGLES
EQUAL ANGLESPROPORTIONAL SIDES
MIDPOINT THEOREM
DE = ½ AB
PARALLEL LINE THEOREM
REFLEXIVE PROPERTY
A=A
SYMMETRIC PROPERTY
IF A=B, THEN B=A
TRANSITIVE PROPERTY
IF A=B AND B=C, THEN A=C
CENTRAL ANGLEOF A CIRCLE
m O = m arc-AB∠
o
B
A
CENTRAL ANGLE
INSCRIBED ANGLEOF A CIRCLE
m A = ½ m arc-BC∠
A
B
C
ANGLE FORMED BY A TANGENT-CHORD
m A = ½ m arc-AC∠A
B
C
ANGLE FORMED BY SECANT-SECANT
m A = ½ [ m arc-BC − m arc-DE ]∠
AB
C
D
E
ANGLE FORMED BY SECANT -TANGENT
m A = ½ [ m arc-CD − m arc-BD ]∠
A
B
C
D
ANGLE FORMED BY TANGENT-TANGENT
m A = ½ [ m arc-BDC − m arc-BC ]∠
A B
CD
ANGLE FORMED BY 2-CHORDS
m 1 = ½ [ m arc-AC + m arc-BD ]∠
A
B
C
C
1