1st Geometry Journal

By Daniel Escobar

What are points, lines, and segments?

• Point: A dot in space that indicates something or a location. Pic: .

• Line: A straight conection of dots that go for ever. Pic:

• Segment: A piece of line that has a begining and an end.

What are rays and Planes

• Plane: a flat surface Pic:• Rays: A conection of dots that have one

begining and go on for ever• How are a line, a segment, and a ray related to

each other?• 1. All of them make shapes.

What is the difference between a collinear and a coplanar point

• Collinear point: Points that lie in the same line.Collinear = line

• Coplanar Point: Points that lie in the same plane.

Coplanar = Plane

Noncollinear vs Noncoplanar

• Noncollinear: points not on the same line

• Noncoplanar

What is an intersection?

• An intersection is the set of all points that two or more figures have in common. My def: when two lines cross each other.

• Pic:

What is the difference between a postulate and an axiom, and theorem?

• Postulate/axiom is a statement that is accepted as true without proof.

• A Theorem is a statement that you can prove.• If you have proven a theorem, you can use it

as a reason in later proofs.

What is a Ruler Postulate

• Ex.1

• Ex.2

• Ex.3

1

A ruler postulate tells us that the points on a line can be paired on a one-to-one with a real number

7

13

2

12

11 14

6 9

What is a Segment Addition Postulate?

• The Segment Addition Postulate states that if B is between A and C then

AB + BC =AC

Ex.1 DF + FG = DG

Ex.2 GT + TE = GE

a b c

D FG

G T E

How to find the distance between two points on a coordinate plane?

• A coordinate Plane is a plane that is divided into four regions by a horizontal line (x-axis) and a verticle line (y-axis). The Distance between two points (x1, y1) and (x2, y2) is found with the following formula, the distance formula:

• d=√¯¯¯¯¯¯¯¯¯Ex.1 d=√¯¯¯¯¯¯¯¯ = √¯¯¯29Ex.2 d=√¯¯¯¯¯¯¯¯¯ =√¯¯¯5 .4Ex.3 d=√¯¯¯¯¯¯¯¯¯ =√¯¯¯15

(x2-x1)^2 + (y2-y1)^2

(1- -4)^2 + (-2-0)^2

(-4-1)^2 + (-4- -2)^2

(-2- -2)^2 + (-8-7)^2

Congruence vs. Equality

• When something is congruent, it is exactly the same/ same measure. Ex: AB≅CD

• When something is equal, it means it has the same value. Ex: 2=2

• They both compare two numbers/ solutions/products.

• Both relate to having two same products.

What Is the Pythagorean Theorem?

• The Pythagorean Thereom states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. a2 + b2 = c2

6

4

62 +42 = 5252 = c2

√52= c7.2 = c

5

5

52 + 52 = 5050 = c2

√50= c7.07 = c

2

92 + 22 = 8585 = c2

√85= c9.2 = c

9

Angles

• An angle are two rays that share a common endpoint. An angle is measured by 3 points. Ex: ABC. The letter in the middle is the ∠vertex.

• There are 4 types of angles • Right angle: Measures 90°• Acute angle: Less than 90°• Obtuse angle: more than 90°• Straight Angle: Measures 180°

What is an Angle addition postulate?

• The Angle addition postulate states that if S is I the interior of PQS + m SQR = PQR.∠ ∠ ∠

Ex.1 m= JKM if m JKL= 42° and ∠ ∠m LKM=28°= 14°∠

Ex.2 m DEG=37° and m DEF=84° find m GEF ∠ ∠ ∠(84-37)= 47°= GEF∠

Ex.3 m LKM if m JKL =56.4° and m JKM ∠ ∠ ∠=82.5° (82.5- 56.4) = 26.1= m LKM∠

Midpoint

• A midpoint is a point that bisects, or divides a segment into 2 congruent segments. It can be found by dividing the measurements of the segments by two. Ex: AB=8 then AM=4, BM=4. AM=BM.

• CD= 5 CM=2.5, DM=2.5 • AB=6 AM=3, BM=3• AC, AB =2y and BC 8y-3• 2y= 8y-3 -8y -8y

-6y=-3 y=2

AB=4, BC= 13, AC = 17

Angle Bisector

• An Angle bisector is a ray that divides an angle into 2 congruent angles.

• To construct one you will need a compass. And follow the instructions on the picture below.

What are adjacent, vertical and linear pairs of angles??

• Adjacent angles: Two Angles that have the same vertex and share side.

• Linear Pairs: are two angles that create a straight line

• Vertical Angle: are two nonadjacent angles formed by two intersecting lines.

Complementary vs. Supplementary

• Complementary angles: are two angles that add up to 90°

• Supplementary angles: are any two angles that add up to 180°

52° 38°

55° 125°

How to find perimeter and area of te following shapes.

• Rectangle: P=2L +2w, A =Lw Ex: L=2cm w= 5cm (P=2 +25) P=27cm, (A=2x5) A=10 cm2

• Triangle: P= a+b+c, A= 1/2bh Ex: a=8, b=(x+1), c= 4x, and h=6 . P=5x +9, A= 3x+3

• Square: P= 4s, A=s2 Ex: 10 cm(P= 10x4) P=40 cm, (A=102) A=100 cm

How to find the area and circumference of a circle

• Area of a circle: A=(pie)2 Ex: 4cm4(pie)2 =16x(pie) ≈50.24 cm2

8cm ≈ 67.14 cm2

Circumference of a Circle: C =2(pie)r (R=radius:

a segment of a circle one of its endpoints are the center of the circle and another point on the circle.)

Ex: 4 cm C=8(pie) ≈25.12 C= 16(pie) ≈50.24

5 step process

• 1st Read and Analyze the Question• 2nd Find important info. And rewrite it• 3rd Visualize the information you just wrote• 4th Solve the equation• 5th Write the answer

Transformations

• Transformation: Change of position of an object.• Image A transformA A prime∆ABC ∆A’B’C’There are 3 types of transformations:TranslationReflection Rotation

Translation

• • • A

• B C• • A’

B’C’

Translation slides an object in any direction.

Reflection

• Reflection reflects/mirrors a figure across the line

If across the x axis (x,y) -> (x,-y). If across y axis (x,y) -> (-x,y)

prime

Rotation

• Rotation is when you rotate a figure around a point.

Prime

Thank you for your patience…