Chapter3 Geometry

3 terms to describe Polygons (in relations to sides and angles)

-Equiangular (angles are equal, like rectangle) -Equilateral (sides have equal side length, like a rhombus) -Regular (all sides and angles are equal to each other, like a square)

Rectangle Properties (regarding symmetry)

1. 2-fold rotational symmetry, at 180d and 360d 2. Axes of symmetry through the midpoints of opposite sides.

Square Properties (symmetry)

1. 4 fold rotational-symmetry (coincides at each 90d) 2. 4 axes of symmetry (2 through midpoint, 2 through vertex)

Parallelogram Properties (regarding symmetry)

1. A parallelogram has no reflectional symmetry. 2. A parallelogram has 2-fold rotational symmetry (at 180d and 360d)

Rhombus Properties (regarding symmetry)

1. Diagonals are axes of symmetry. 2. Two-fold rotational symmetry - at 180d, 360d

Two aspects of, for example, a 4 fold rotational symmetry

1. I could turn it, and 4 times it would coincide. -----at 90 degrees, 180 degrees, 270 degrees, and 360 degrees the shape would coincide 2. The smallest angle to have it coincide is thus 360/4 = 90 degrees

Kite Properties

1. Long diagonal bisects shorter diagonal. 2. Top two angles (including leg of diagonal) are congruent to each other. 3. both opposite sides are congruent to each other 4. 1 axis of symmetry through the middle, no rotational symmetry.

Trapezoid Properties (regular)

1. Same side angles are equal 2. Opposite non-parallel sides have equal measure 3. Diagonals are congruent, but not perpendicular. 4. has a axis of symmetry down the middle.

How to write a proof for consecutive interior angles theorem?

1. Use fact that lines are parallel, as a given statement. 2. Use corresponding angles post, then linear pair postulate, then substitution, and define supplementary. -----don't use vertical angles (for congruence) when you're talking about supplementary angles.

How do you prove the alt int angles theorem, and alt ext angles theorem?

1. Use parallel lines as a given statement. 2. Use corresponding Ang postulate, and vertical angles theorem, then connect the two with transitive property for CONGRUENCE.

Trapezoid Properties (irregular)

1. is a quadrilateral but not a parallelogram. 2. Same side angles (which can exist even without parallel lines) add to 90 degrees in a trapezoid. 3. In a Rectangle Trapezoid / Right Trapezoid, the base and shorter leg have equal measure. 4. Seems to have no reflectional or rotational symmetry. 5. Diagonals are not congruent. nor perpendicular

Defining polygons.

1. straight sides are segments 2. number of sides = number of angles 3. flat 4. closed 5. nothing inside

Practice... An equilateral triangle has 3-fold symmetry, so it will coincide with itself after a rotation of...

360/3, or 120d rotation. Use a circle and divide into n parts in order to rotate a figure.

Square properties (conditional)

All conjectures apply (from parallelogram), --- It's a regular quadrilateral, so yes. --- rules of diagonals also apply (bisect each other, are angle bisectors). 1. Is a type of parallelogram, square, and rhombus. (look back to Euler Diagram!) 2. is a regular sided figure

Exterior Angle of Polygons

An angle that forms a linear pair with an angle of a polygon

Triangle Symmetry Conjecture

An axis of symmetry in a triangle is the segment bisector of the side it intersects, and it passes through the vertex of the angle opposite that side of the triangle.

Remote Interior Angles

Angles of the polygon for an exterior angle of the polygon at a given vertex. (each exterior angle has two remote interior angles)

Use a few steps to write proofs for all of these converses! How?

First three steps relate congruence of three equations, then employ the transitive property in these 3 equations. Then, by the "converse of xxx theorem," you prove that the lines are parallel. ----WHEN PROVING A THEOREM, don't use that theorem to prove itself.

For reflection and rotational symmetry in Reg Polygons...

For reflection and rotational symmetry in Reg Polygons...

Lesser Circle

Forms by cutting the sphere anywhere but the middle part (egg slicer)

Great Circle

Forms by intersecting a sphere with a plane through the center. (Orange slice)

The Parallel Postule

Given a line and a point not on the line, there is exactly one line through the point that is parallel to the given line.

How to Prove the Exterior Angle Theorem

Given the shape and exterior angle, use linear pair postulate (180d), triangle sum theorem (180d) , transitive, and then subtraction.

How to write a proof for the converse for the consecutive angle theorem (supplementary).

Given, defin. of supplementary, compare to a third angle and use the linear pair postulate. transitive, then subtraction, then how you know its plural. 3. the third angle should be a corresponding angle of 1 given angle, and a Linear Pair Supplement for the 2nd given angle. -----don't use vertical angles (for congruence) when you're talking about supplementary angles.

Axis of symmetry in an Isosceles triangle.

Goes through the vertex angle and through the midpoint of the base side.

Polygons sides names

Heptagon (7), Octagon (8), Nonagon (9), Decagon (10), 11-gon, Dodecagon (12), 13-gon, n-gon

Quick Concept / Reflectional -

How many axes + where they are {reflectional}

Quick Concept / Rotational

How many folds + how many degrees {rotational}

Converse for consecutive angle theorem (supplementary).

If 2 lines are intersected by transversal so that consecutive interior angles formed are supplementary, then the lines are parallel.

Converse of alternate exterior angles theorem theorem

If 2 lines are intersection by a transversal so that the alternate exterior angles formed are congruent, then the lines are parallel.

Converse of alternate interior angles theorem.

If 2 lines are intersection by a transversal so that the alternate interior angles formed are congruent, then the lines are parallel.

Converse of corresponding angles postulate.

If 2 lines intersected by a transversal from congruent corresponding angles, then the lines are parallel.

Rotational Symmetry definition

If a figure has at least one rotation image, not counting rotation images of 0 degrees or multiples of 360 degrees, then it coincides with the original image.

N-Fold Rotational symmetry

If a figure has n-fold rotational symmetry, then it will coincide with itself after a rotation of (360/n). For example, an equilateral triangle has 3 fold symmetry, so it will coincide with itself after a rotation of (360/3) = 120.

Reflectional Symmetry

If a figure's reflected image coincides exactly with the preimage, then t has reflectional symmetry. -line: axis of symmetry.

Parallelogram Properties!! (Conditional)

If a quadrilateral is a parallelogram, then... 1. Opposite angles are congruent. 2. Opposite sides are congruent. 3. The Diagonals bisect each other. 4. The consecutive angles are supplementary.

Rhombus Properties (conditional)

If a quadrilateral is a rhombus, 1. rhombus is a type of parallelogram 2. Diagonals are perpendicular to each other. 3. The diagonals are angle bisectors. (which also bisect each other.)

Rectangle Properties (conditional)

If a quadrilateral is rectangle, then. 1. It's a type of parallelogram 2. Diagonals are congruent.

Corresponding Angles Postulate

If parallel lines are intersected by a transversal, then corresponding angles are congruent.

Alternate Exterior Angles theorem

If parallel lines are intersected by a transversal, then the alternate exterior angles formed are congruent.

Alternate Interior Angles Theorem

If parallel lines are intersected by a transversal, then the alternate interior angles are congruent.

Consecutive (same side) Interior Angles

If parallel lines are intersected by a transversal, then the consecutive interior angles are supplementary.

Rotational Symmetry procedure

If rotated, the figure will coincide with its original shape before it is turned 360 degrees

(3.2) Properties of Quadrilaterals

In the lesson, we will make conjectures about special quadrilaterals, then prove them. -we use the definitions of these quadrilaterals!

(3.5) Triangle Sum Theorem

Inner angles of triangle add to 180d. TRY! - Rip off edges and align to make a line.

Central Angle of Polygon

Innermost angle of one of the sections/orders of regular polygon.

(3.3) Parallel Lines and transversals! What are transversals?

Line, segment, or ray that intersects 2 or more coplanar lines, segments, or rays, each at a different point. ----DOES NOT require the lines to be parallel!

Definition of parallel lines

Lines which are coplanar and don't intersect.

Corresponding Angles

Lines which share a transversal, the angles at the same relative position orf the four angle pairs formed each.

Consecutive (same side) Interiors

Located on the SAME SIDE of the transversal, inside of the two parallel lines.

Alternate Interior Angles

Opposite sides of transversal, located inside of the two lines (parallel)

Alternate Exterior Angles

Opposite sides of transversal, located outside of the two lines (parallel)

Antipodal Points

Points opposite on sides of sphere (poles)

(3.1) Symmetry in Polygons - 2 concepts

Reflectional, Rotational

When solving for degree measures of a triangle using algebra, what should you do?

Set sum equal to 180d, Combine like terms, set the quadratic equal to zero, factor, pull out two solutions, eliminate extraneous solutions.

vs. Planar Geometry

Surface - Plane Line.

Spherical Geometry!

Surface - Sphere Great circles are the longitude (like an atom)

Existence / Uniqueness Postulate - Describes the Overall TYPE

Tells us there's only ONE of something -exactly one line, exactly one point, etc.

What's the key to all of these angle postulates and theorems?

That they use the fact that lines are parallel, then make a claim of angles.

What's the key to writing all of these converse of theorems/postulates?

That you start with knowledge of congruence or supplementary, and prove that two lines are parallel.

Exterior Angle Theorem

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

What are extraneous solutions in terms of the measures and distances in math?

They all solve the equation, but they don't make sense in the context of the problem, or the rules of geometry.

Polygons can be equiangular and not equilateral, or vice versa.

This all depends on its side/angle measures.

(3.4) How do we establish the measures of angles without using the rule that every polygon = 180d?

Use past rules. -angle addition postulate -corresponding angle postulate.

In a triangle with a line parallel to its base, how do we determine which angles (out of 5) are congruent?

Use the linear pair postulate, and addition of angles. -When writing proofs, you will be given parallel lines and must prove that the three angle measures sum to 180d.

Odd number of sides + vertex

Vertex to midpoint = n

Even number of sides and vertex

Vertex to vertex = n/2 Vertex to midpoint = n/2 Therefore adding to 1 whole, or n

Some nonregular figures which have some symmetry are...

an isosceles trapezoid, and a kite.

The side that's not a leg is the...

base

Orthographic

describes the view of parallel lines, where the plane is perpendicular to the view (bird's eye view of a train track)

smallest angle for reflection and rotational symmetry in Reg Polygons...

equals 360/n (the rest are multiples to 360)

Three congruent sides=

equilateral

A type of isosceles is an....

equilateral (In euler diagram, equilateral is the hypothesis/inside, while isosceles is the conclusion/outside)

Isoceles Trapezoids

http://www.coolmath.com/reference/trapezoids

At least two congruent sides

isosceles

Two congruent sides of a triangle are called....

legs

N number of axes of symmetry are found for a shape with..

n sides

Turning a figure 0 degree or 360 degrees is...

not really rotational symmetry at all (just like 0 or 1 fold symmetry, not real)

no congruent sides

scalene

Drawing a circle around the polygon and dividing the circle equally to make the regular polygon could also give you...

the center of mass.

In regular polygons, intersecting axes of symmetry give you...

the center of mass.

Coincide means...

to overlap/align perfectly when rotated around a center point

Parts of a triangle (5)

vertex angle, legs, axis of symmetry, base, base angles.

Rotation symmetry means it coincides x times, not that it necessarily has...

x number of sides.