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Theorem 6.1

Two lines parallel to a third line are parallel

Given a transversal

two parallel lines will create 8 angles

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Theorem 6.2

let the lines l and l' be cut by a transversal t. If the sum of the measures of two interior angles on the same side of t is 180° or if two alternate interior angles are congruent, then the lines l and l' are parallel.

Corollary from 6.2

If two lines are perpendicular to a third line, then these lines are parallel

Theorem 6.3

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

Theorem 6.4

(Angle sum theory) If two parallel lines are cut by a transversal, any two alternate interior angles are congruent

Quadrilateral

Let A,B,C and D be four points, no three of which are on the same line. the union of four segments, AB, BC, CD, and DA is called a quad if no two of these segments have a common point other than a point from {A, B, C, D}

Diagonals of a quadrilateral

are segments AC and BD. Vertices A and C (B and D) are opposite of each other

Convex

a quadrilateral whose diagonals intersect

parallelogram

a quadrilateral whose two pairs of opposite sides are parallel.

AB || CD, and BC || AD

Theorem 7.1

Let ABCD be a given quadrilateral. If the intersection point of its diagonals bisects each of the diagonals, then ABCD is a parallelogram.

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Theorem 7.2

In every parallelogram opposite sides are congruent and opposite angles are congruent

Theorem 7.3

If ABCD is a parallelogram then its diagonals bisect each other

Theorem 7.4

the line l passing through the intersection point O of the diagonals of the parallelogram ABCD, intersects the sides AB and CD at E and F respectively. Show that OE = OF

Theorem 7.5

1) If two of the sides of a given quadrilateral are congruent and parallel, then this quad. is a parallelogram

2) if the opposite sides of a quad ABCD are congruent, then this quad is a parallelogram

Theorem 7.6

Diagonals in every rectangle are congruent

Theorem 7.7

In every rhombus, diagonals are angle bisectors and they are perpendicular to each other.

Rectangle

a parallelogram in which all angles are right angles

Square

a rectangle in which all sides are congruent

Rhombus

parallelogram in which all sides are congruent

Thales' Theorem

Three parallel lines intersect the sides of a given angle. If the two segments cut by these lines on one side of the angle are congruent, then the segments cut by these lines on the other side of the angle are also congruent

Theorem 8.1

Two parallel lines, intersecting the sides of a given angle, cut proportional pairs of segments on the sides of this angle.

Theorem 8.2

If two right triangles ABC and A'B'C' are such that α = ∠A congruent to ∠A' and m∠C = m∠C' = 90° then cosα = cosα'

Theorem 8.3 (Pythagorean theorem)

In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.

Corollary 8.4 (consequence of the theorem)

i) In any right triangle, the lengths of the hypotenuse is greater than the length of any of the legs

ii) For any acute angle α, 0<cosα<1

Theorem 8.5

(Triangle inequality) for any three points A, B, and C in the plane AB + BC ≥ AC

Theorem 8.6 (converse of pythagorean theorem)

if a triangle with side lengths a, b, and c is such that a^{2}+b^{2}=c^{2}, then this triangle is right, with the right angle opposite the side with length c.

Circle

a set of points in the plane equidistant from a given point

Radius

every segment with one of its endpoints on the circle and the other endpoint the center of the circle

chord

a segment with its endpoints on the circle

diameter

a chord that contains the center of the circle

Theorem 11.1

For any three points not lying on the same line, there exists unique circle passing through these three points

theorem 11.2

a tangent line to a given circle has only one point in common with the circle

a circle is inscribed in a triangle if

every side of the triangle is tangent to the circle

theorem 11.3

the three angle bisectors of a given triangle intersect in a single point. this point is the center of the circle inscribed in this triangle

transversal

Given two parallel lines, any line t intersecting both lines l and l' and not passing through their intersection point

tangent line

the line passing through a point on the circle and being perpendicular to the radius of the circle at that point

Cosine of an angle

in a right triangle, the ratio of the side adjacent to the angle and the hypotenuse

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