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

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)

(x, y) (-x, -y)Same as a reflection in the

origin

(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

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

SUPPLEMENTARY ANGLES

Documents

Transcript of SUPPLEMENTARY ANGLES