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- Arizona
- Arizona State University - Tempe
- Mathematics
- Mathematics 310
- Roh
- mat 310 defintition 1.1-3.5

Saul H.

Parallel Postulate

Given a line L in a plane and a point P which is in the plane but is not on the line L, there is exactly one line in the plane passing through P which is parallel to L.

Rigid Motion

for each point P of the plane, F assigns a point F(P) to P satisfying the distance preserving property, that is, for any two points A and B, the distance between A and B is equal to the distance between F(A) and F(B)

- |AB| = |F(A)F(B)|.

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ASA

Given two triangles △ABC and △A0B0C0 so that |AB| = |A0B0|, |∠A| = |∠A0|and|∠B| = |∠B0|, △ABC ∼ = △A0B0C0

SAS

Given two triangles ABC and A0B0C0 so that |∠A| = |∠A0|, |AB| = |A0B0|, and |AC| = |A0C0|, then △ABC ∼ = △A0B0C0.

SSS

Two triangles with three pairs of equal sides are congruent.

HL

If two right triangles have equal hypotenuses and one pair of equal legs, then they are congruent.

Assumptions about Transformations

- map lines to lines, rays to rays, and segments to segments.
- are distance-preserving, the distance between the images of two points is always equal to the distance between the original two points.

- are degree-preserving, the degree of the image of an angle is always equal to the degree of the original angle.

Congruence

A congruence in a plane is a transformation of the plane which is equal to the composition of a ﬁnite number of basic rigid motions. We also say that for two subsets S and S′ of a plane, S is congruent to S′, and we write S ∼ = S′, if there is a composition of a ﬁnite number of basic rigid motions

Congruent Triangles

Let △ABC and △A0B0C0 be any triangles in a plane. Write △ABC ∼ = △A0B0C0 if and only if there is a congruence F that maps △ABC onto △A0B0C0 and F(A) = A0, F(B) = B0, F(C) = C0.

Transformation

F is a transformation of a plane if and only if for any point P in a plane, there exists a unique point Q in the plane such that Q = F(P).

Angle

An angle is the union of two rays with a common vertex. The angle formed by the two rays ROA and ROB is denoted by ∠AOB.

- collinear = straight angle
- coincide = zero angle
- Two angles are equal or congruent if they have the same degree.

Lines and Segments

A line is a straight line indeﬁnite in both directions.

Two lines are distinct if there is at least one point that belongs to one but not the other.

Two lines are the same if they are not distinct.

Two lines are parallel if they have no point in common. denoted by L1||L2

Degree

The degree is a unit of angle measure deﬁned such that an angle of one degree corresponds to the 1/360 of the full rotation around a ﬁxed point

Adjacent Angles

Two angles ∠AOC and ∠COB are said to be adjacent if they have a side in common and if C lies in the convex part of ∠AOB.

Perpendicular lines

Two lines (segments) are perpendicular if an angle formed by the two lines at the point of intersection is a right angle. LOA⊥LOB and OA⊥OB are notations of perpendicular lines and perpendicular line segments.

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Perpendicular Bisector

For any segment AB, the line perpendicular to LAB and passing through the midpoint of AB is called the perpendicular bisector of AB

Angle Bisector

A ray ROC in the convex part of an angle ∠AOB is called an angle bisector of ∠AOB if |∠AOC| = |∠COB|.

Midpoint

For any segment AB, the point C in AB so that |AC| = |CB| is called the midpoint of AB.

Convex set

A subset R in a plane is called convex if given any two points A, B in R, the segment AB lies completely in R.

Segment

If A and B are two distinct points, then by (A1), there is a unique line containing A and B. We denote this line by LAB and call it the line joining A and B. On the line LAB, we denote by AB the collection of all the points between A and B in LAB together with the points A and B themselves.

Ray

The set of the point P and the points from a half-line, L+ or L−, is called a ray. We also say that these are rays issuing from P. We use the symbol RPA to speciﬁcally refer to the ray from P to A. The point P is called the vertex of the ray RPA.

Rotation

The rotation Ro of t degree (−180 ≤ t ≤ 180) around a given point O, called the center of the rotation, assigns each point P, to a point Ro(P) of a plane

- If t = 0 or P = O, then Ro(P) = P
- If 0 < t ≤ 180 and P ̸= O, = counter clockwise
- If −180 ≤ t < 0 and P ̸= O, = clockwise

Reﬂection

The reﬂection R across a given line L, where L is called the line of reﬂection, assigns to each point on L the point itself, and to any point P not on L, R assigns the point R(P) which is symmetric to it with respect to L, in the sense that L is the perpendicular bisector of the segment joining P to R(P).

Quadrilateral

A quadrilateral is deﬁned as a geometric ﬁgure consisting of the four distinct points A, B, C, and D in a plane, together with the four segments AB, BC, CD, and DA so that none of these segments intersects any other except at the end points as indicated. In symbol, the quadrilateral is denoted by ABCD.

Rectangle

A quadrilateral ABCD is called a rectangle if all of whose angles are right angles, i.e., |∠A| = |∠B| = |∠C| = |∠D| = 90◦

Parallelogram

A quadrilateral ABCD is called a parallelogram if LAB||LCD and LAD||LBC.

- For any rectangle ABCD in a plane, ABCD is a parallelogram

Equal Segments

For any segments AB and CD, AB is said to be equal (congruent) to CD if |AB| = |CD|

Equal Angles

For any angles ∠ABC and ∠DEF, ∠ABC is said to be equal (congruent) to ∠DEF, and denoted by ∠ABC ∼ = ∠DEF if |∠ABC| = |∠DEF|

Vector

A vector is just a segment together with the designation of one of its two endpoints as a starting point; the other endpoint will be referred to as the end point of the vector. The length of a vector is deﬁned as the length of the segment joining the starting point and the endpoint of the vector.

Equal Transformations

Two transformations F1 and F2 are said to be equal, in symbol, F1 = F2, if for every point P, F1(P) = F2(P)

Equal Vectors

Two vectors −→ A1B1 and −→ A2B2 are said to be equal, and denoted as −→ A1B1 = −→ A2B2 in symbol, if LA1B1||LA2B2, |A1B1| = |A2B2| and B1 and B2 are on the same side of the line LA1A2.

equal or congruent angles

Two angles ∠ABC and ∠DEF are said to be equal or congruent if |∠ABC| = |∠DEF|.

Identity

The identity is a transformation that leaves every point unchanged, denoted by I.

I(P) = P

Inverse Transformation

For any transformation F of a plane, a transformation G is called an inverse transformation of F if F ◦G = G◦F = I

Angles and Sides of Isosceles Triangles

If △ABC is an isosceles triangle with |AB| = |AC|, then ∠B and ∠C are called its base angles, ∠A is called its top angle, and the side BC is called its base.

Median of a triangle

For any side of a triangle, the median of the side is the line joining the midpoint of the side of the triangle to the opposite vertex.

Altitude of a Triangle

For any side of a triangle, the altitude on the side is the line passing through the opposite vertex and perpendicular to the side. Sometimes the segment from the vertex to the point of intersection of this line with the line containing the side is called altitude.

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