1-3 Points, Lines, Planes

plane M or plane ABC (name with 3 pts)

A

point A

Points A, B are collinearPoints A, B, and C are coplanar

•Intersection of two distinct lines is a point•Intersection of two distinct planes is a line

1-4 Segments, Rays, Parallel Lines, Planes

Segment AB or

•Opposite rays share same endpoint•Opposite rays are collinear

Parallel Lines

-Never intersect

-Extend in the same directions

-Coplanar

Skew Lines-Never intersect-Extend in different directions-Noncoplanar

Parallel Planes-Can never intersect

1-5 Measuring Segments

AB is the abbreviation for the distance between points A and B.

Segment Addition PostulateMidpoint•B is exactly halfway between A and C•B is the average coordinate of A and C

1-6 Measuring Angles

1st Semester Geometry Notes page 1

vertex

Congruent Anglesm 1 = m 2 (the measure of angle 1 equals the measure of angle 2)

1 ≅ 2 (Angle 1 is congruent to angle 2)(May also be indicated by arc on both angles)

Angle Addition Postulate

mAOB + mBOC = mAOC

Pairs of Angles1 and 2 are adjacent angles-No interior points in common

-Share the same vertex R-Share common side

1 and 32 and 4

Vertical Angles -Non-adjacent; -Formed by two intersecting lines-Are congruent

Compl.Corner

Suppl.Straight

Linear Pairs-Form a straight angle-Are supplementary (sum = 180)

1-1 Inductive ReasoningConditional: If a, then b statementa is the hypothesis; b is the conclusionConverse: Switch the hypothesis and conclusion.If b, then a.Truth value of a statement: Either True or False, where True means always trueBiconditional: Both the conditional and its converse are true. You can combine both statements with if and only if.a if and only if b.

Addition Property

If a = b, then a + c = b + c

Subtraction Property

If a = b, then a – c = b – c

Multiplication Property

If a = b, then a * c = b * c

Division Property

If a = b and c ≠ 0, then a/c = b/c

Reflexive Property

a = a

Symmetric Property

If a = b, then b = a

Transitive Property

If a = b and b = c, then a = c

Substitution Property

If a = b, then b can replace a in any expression

Distributive Property

a(b + c) = ab + ac

2-1 2-2 2-3 5-4 Deductive Reasoning

Law of Detachment: If a, then b. (True)Given: a is Trueb is therefore True.

Law of Syllogism If a, then b. (True)If b, then c. (True)If a, then c must be True.

A BA B C

If A is falls to the right, then B falls to the right

Given: A falls to the right is TrueThen: B falls to the right.

If B is falls to the right, then C falls to the right.If A falls to the right, then C falls to the right.

2-4 Algebraic Properties

1st Semester Geometry Notes page 2

1-8 8-1 Pythagorean Theorem, Midpoint, Distance Formula

(leg1)2 + (leg2)2 = hypotenuse2

True only for right triangles

Pythagorean Theorem

1st Semester Geometry Notes page 3

Classifying TrianglesLet a, b, c be the lengths of

the sides of a triangle, where c is the longest

Acute: c2 < a2 + b2

Right: c2 = a2 + b2

Obtuse: c2 > a2 + b2

Distance between 2 points-Use the Pythag. Thm Midpoint

(average x, average y)

Transversal: line that cuts across two or more lines

Congruent if and only if l and m are parallel

Supplementary if and only if l and m are parallel

Same-side exterior:∠1 and ∠4∠5 and ∠8

3-1 3-2 3-3 Parallel Lines and Angles

Vertical angles are congruentLinear pairs are supplementary

3-4 Triangle Sum Thm, Exterior Angle Thm

A triangle is isosceles if and only if the base angles are congruent.

A triangle is equilateral if and only if thetriangle is equiangular

4-5 Isosceles and Equilateral Triangles

1st Semester Geometry Notes page 4

180 – n360

3-5 Polygon AngleSum Thmsn = number of sides

Interiorangle

Exteriorangle

for regular polygons

3-6 3-7 Graphing Equations of Lines

•Any point on the line must satisfy the equation of the line (y = mx + b)•Parallel lines have equal slopes (same steepness)•Perpendicular lines have slopesthat are negative reciprocals of each other

Standard Form:Ax + By = CPoint-Slope Form:y – y1 = m (x – x1)

9-1 Translations:

(x, y) → (x+a, y+b)

9-2 Reflections:Preimage and image are -on opposite sides of line of reflect.-equidistant from line of reflectionReflect about x-axis

(x, y) → (x, -y)Reflect about y-axis

(x, y) → (-x, y)Reflect about y = x

(x, y) → (y, x)

9-3 Rotations about origin:For each 90° of rotation,

switch the x and y coordinates; then determine signs based on the quadrant after rotation

9-4 Symmetry

Preimage: before the transformationImage: after the transformationIsometry: size and shape stay the sameReflections, Translations, and Rotations are isometries

9-5 Dilations

Enlargement:Multiply both x and y by a scale factor k greater than 1(x, y) → (kx, ky)Reduction:Multiply both x and y by a scale factor k between 0 and 1(x, y) → (kx, ky)Vertical stretchMultiply the y only by a scale factor k greater than 1(x, y) → (x, ky) Horizontal shrinkageMultiply the x only by a scale factor k between 0 and 1(x, y) → (kx, y)

1st Semester Geometry Notes page 57-1 Ratios and Proportions

7-2 Similarity

7-3 Proving Triangles Similarity SSS, SAS, AA

=42

63

Small leg Medium leg Hypotenuse

∆1 a b c

∆2 r h a

∆3 h s b

1) Redraw and label triangles.2) Fill in the table with given information3) Use proportions or Pythag. Thm to

solve for missing lengths

1st Semester Geometry Notes page 67-4 Similarity in Right Triangles

7-5 Proportions in Triangles

in a triangle

4-1 Congruent Polygons

Two polygons are congruent if they have the same size and shape.

Two polygons are congruent if and only if all corresponding sides and corresponding angles are congruent.

1st Semester Geometry Notes page 74-2 4-3 4-6 Proving Triangles Congruent SSS SAS ASA AAS HL

CPCTC is an abbreviation of the phrase “Corresponding Parts of Congruent Triangles are Congruent.”

4-4 4-7 Using CPCTC in Proofs

1st Semester Geometry Notes page 8

5-5 Triangle Inequalities

5-1 5-2 5-3 Special Segments in Triangles

Altitudes (vertex to opposite side at right angles – Orthocenter

Perpendicular Bisectors (90 degrees through midpoint of a side) – Circumcenter

Medians (vertex to midpoint of opposite side) – Centroid Angle Bisectors (vertex to opposite

side through line splitting the vertex angle in half– Incenter

Given two sides a and b, the third side the triangle with length c must satisfy | a – b | < c < a + b