Distance Formula

or

Length of a segment

having endpoints

(x₁ , y₁ ) and ( x₂ , y₂ )

adjacent supplementary angles

[image]

1.  has endpoints which are the midpoints of two sides

2.  is parallel to the third side of the triangle

3.  has a length half of the third side

2(length of midsegment) = length of third side

The measure of the exterior angle of a triangle is equal to the sum of the two remote interior angles.

[image]

perpendicular

_l_

intersect to form right angles

equal slopes

never intersect

coplanar

Any value is equal to itself.

Any figure is congruent to it.

3=3

AB≅AB

If two sides of a triangle are known, how do you find the possible range of values for the third side?

Let two sides be x and y.

x>y

How can you determine if three segments lengths can form a triangle?

Let the three sides be x,y, and z.

The smallest values are x and y.

If a=b then b=a

If m<ABC= 40°, then 40° = m<ABC.

Midpoint of a segment having endpoints

(x₁, y₁)(x₂, y₂)

x₁ +x₂    ,    y₁ + y₂

   2                 2    

Slope "m" of the line containing two given points

(x₁, y₁ ) and (x₂, Y₂)

y₂ -y₁

x₂-x₁

Parallel lines are coplanar and do not intersect.

Parallel lines have same slope "m" in the equations

y=mx+b

and y-y₁ = m(x-x₁)

The slopes of perpendicular lines are opposite reciprocals.

If one is 2/3, the other is  -3/2

The product of the slopes of perpendicular lines = -1

2 • -3 = -1

 3     2        

If p, then q

p→q

If q, then p

q→p

  ̴q   →       ̴p

The contrapositive has the same truth value as the original conditional.

p→ q, q→ r, p→ r

If p, then q.  If q, then r. 

∴ If p, then r.

p→q

p

_______

∴q

It does not work backwards.

p→q

q

_______

no conclusion possible

two rays with the same endpoint which are collinear

<----•-----------------•------------------•---->

A                      B                        C

ray BA  and ray BC are opposite rays.

if two parallel lines are cut by a transversal, the alternate interior angles are congruent.

[image]

m∠5=m∠6         and       m∠7= m∠8    

If two parallel lines are cut by a transversal, the same-side interior angles are supplementry and have a sum of 180°.    

[image]

If two parallel lines are cut by a transversal the corresponding angles are congruent (equal). [image]

[image]

List the sides of the triangle in increasing order.

[image]

List the angles in decreasing order.

[image]

What is the possible range of values for s?

7-4 < s < 7+4

3<s<11

If two parallel lines are cut by a transversal, same-side exterior angles are supplementary.

[image]

a+b=180

[image]

If two parallel lines are cut by a transversal, alternate interior angles are congruent.

<1=<8  and  <2=<7

The intersection of the three medians of a triangle.

[image]

1. used to find the length of a segment on a number line

2.  if the endpoints are A and B

the length of AB = l A-B l

3.  Absolute value will produce a positive answer.

share a common side and a common vertex, but no common interior points.

[image]

http://www.hstutorials.net/math/geometry/constructions/perpendicular_not_on_line.htm

 [image]

p  ↔  q

p if and only if q

180(n-2)

or

180(#sides - 2)

360/#sides

or

360÷sides

360/ext. angle

or

360÷(measure of one exterior angle)

Supplementary

add up to equal 180

Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of one is less than the included angle of the other, then the third side of the first triangle is shorter than the third side of the second triangle.

perpendicular

and

bisect opposite angles

[image]

A parallelogram with 4 congruent sides.

The diagonals of a rhombus are perpendicular.

The diagonals of a rhombus bisect the opposite angles.

[image]

MN is the midsegment. 

 M and N are the midpoints of the legs.

MN is parallel to AD and BC.

MN =½(AD + BC)

formula to find arc length in terms of [image], if you know the degree measure of the central angle m° and radius r of the intercepted arc.

m° ( 2 [image] r ) 

360           

m° ( [image] r² )

360