Point A location in geometry. A point has no dimensions without any length, width, or depth. This is represented by a dot and is usually labelled.

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1 Test Date: November 3, 2016 Format: Scored out of 100 points. 8 Multiple Choice (40) / 8 Short Response (60) Topics: Points, Angles, Linear Objects, and Planes Recognizing the steps and procedures for constructions Equations resulting from congruent line segments Make basic arguments to prove your constructions are valid Definitions: Point A location in geometry. A point has no dimensions without any length, width, or depth. This is represented by a dot and is usually labelled. Line A one dimensional, linear object in geometry. A line is defined by a line of points that extends infinitely in both directions. Lines may have length, but no width or depth. Lines can be made by connecting any two points and drawing arrows to extend that line. The line is labeled by both end points with a line on top. For example: AB. Shape A two dimensional, linear object in geometry. Shapes can include circles, triangles, rectangles, etc. Regular polygons have the same number of vertices and sides. The dimensions of a shape are measured by the length and width. Area can be found by plugging these two variables into a certain equation. For perimeter, simply add the lengths of all the sides together. Planes A two dimensional surface made up of infinite points extending infinitely on a surface. A plane has infinite length and width, but no depth. When two planes intersect, they form a line with collinear points naming it. Planes are named for three points that are on the same plane. Collinear Points are considered to be collinear if and when they all lay on a single straight line. When related to geometry, points that are collinear may also be called an axis. For example: Axis of Symmetry. Coplanar When points are on the same plane, they are coplanar. Three points that are coplanar can name a plane. For example: Plane XYZ. Angle The measure of space (in degrees) between two intersecting lines or planes. The point where the two lines share a common point (intersect) is called the vertex. Minimum Rotation of Regular Polygons to keep symmetry: Triangle (3 sides) Octagon (8 sides) - 45 Square (4 sides) - 90 Nonagon (9 sides) - 40 Pentagon (5 sides) - 72 Decagon (10 sides) - 36 Hexagon (6 sides) - 60 Hendecagon (11 sides) Heptagon (7 sides) Dodecagon (12 sides) - 30 *Rule: The minimum rotation of a regular polygon is the difference between the measure of an interior angle and 180.

2 Constructions: Equilateral Triangle: 1. Start with a line segment labeled with point A and point B. 2. Draw a circle with center A and radius AB. 3. Draw a circle with center B and radius AB. Label intersection C. 4. Connect A, B, and C with line segments Equilateral Triangle Inscribed in Circle: 1. Start with a line segment labeled with point A and point B. 2. Construct a circle with center A and radius AB. 3. Construct a circle with center B and radius AB. Label intersection C. 4. Continue using radius AB to construct 5 more circles. Label intersections appropriately 5. Connect three non-adjacent points (ex. C, E, G) to form a equilateral triangle. Square Inscribed in a Circle: 1. Start with circle and draw diameter AB. 2. Construct the perpendicular bisector of AB; Draw circle A and radius AB, and circle B with radius AB. Label intersections C and D. 3. Connect C and D with line segments. 4. Label the intersections with CD and circle O as E and F. 5. Connect points A, B, E, F to form a square. Hexagon Inscribed in a Circle: 1. Follow instructions above for Equilateral Triangle Inscribed in Circle. 2. Instead of connecting non-adjacent points, connect all points on circle A. Perpendicular Bisectors: 1. Start with line segment AB and construct circles A and B with radius AB. Label intersections C and D. 2. Connect points C and D. Angle Bisectors: 1. Start with angle ABC and construct circle B with arbitrary radius. 2. Label intersections D and E. 3. Construct circle D with radius DE and circle E with radius DE. 4. Label intersection F of circles D and E. 5. Draw ray BF to bisect angle ABC.

3 Copy an Angle: 1. Start with angle ABC and construct circle B with arbitrary radius and label intersections G and H. 2. Draw ray DE (far away) and circle D with radius BG. Label intersection I. 3. Construct circle I with radius GH and label intersection J with circle D. Draw ray DF through point J. Circumcenter: 1. Start with triangle ABC and construct perpendicular bisectors for two of the sides. Label the intersection of the bisectors D. 2. Construct circle D with radius DA. Incenter: 1. Start with triangle ABC and construct angle bisectors for two of the angles. Label the intersections D. 2. Construct circle D with the circle being tangent to each side. Parallel Line Through a Point: 1. Start with Line AB and point C. Construct ray AC. 2. Construct circle A with arbitrary radius and label intersection with line AB, point D. Label intersection with ray AC, point E. 3. Construct circle C with radius AD and label the upper intersection, point F. 4. Construct circle F with radius DE, and label intersection with circle C, point G. 5. Construct line CG, this is parallel to line AB. Axis of Symmetry Given Two Reflected Figures: 1. Draw a line segment between two corresponding points. 2. Construct the perpendicular bisector of that line segment. 3. The bisector is the axis of symmetry. Reflect a Figure Given Axis of Symmetry: 1. Construct circle A with two intersection points on the AOS. Label those points D and E. 2. Using the same radius, construct two circles D and E. 3. The intersection point is the corresponding reflected point. 4. Repeat above steps for all points in the figure.

4 Angles: Supplementary Two or more angles that add up to 180, a straight line. Two congruent angles that are also supplementary are both right angles (90 ). For example: Angle 1 and Angle 2 are supplementary, so are angles 1 and 3. Complementary Two or more angles that add up to 90, a right angle. Two congruent angles that are also complementary are both 45 degree angles. Vertical Angles A pair of intersecting lines that have opposite congruent angles. Vertical angles are always formed with two intersecting lines. For example, Angle 2 and Angle 3 is a pair of vertical angles, so are angles 5 and 8. Alternate Interior Angles that are congruent across the transversal on the same side of parallel lines. For example, Angle 3 and Angle 6 is one pair of alternate interior angles, they are congruent to each other. Alternate Exterior Angles that are congruent across the transversal on opposite sides of the parallel lines. For example, Angle 1 and Angle 8 are alternate exterior, so are Angle 2 and Angle 7. The angles are congruent to each other. Corresponding Angles Angles that are in the same position as the other angle on a parallel line; they are congruent. For example, angles 1 and 5 are congruent because they are corresponding, so are angles 4 and 8. Interior Angle Sum Theorem The interior angles of a triangle always add up to 180 degrees. A common rule for the sum of the interior angles of any polygon is 180(n 2) with n being the number of sides in the polygon. Exterior Angles The supplement of one angle is equal to the sum of the two non-adjacent interior angles. For example, the measurement of angles 2 and 3 are equal to the measurement of angle 4.

5 Proofs and Justifications: When writing proofs, be sure to go step by step until the final statement. Two-column proofs are effective; use a T-chart. One side has the statement; the other side has the reasoning behind it. Always put Givens first then put step by step logic. Do not use the answer as a reason or to prove another statement. Only use what is currently known and do not assume. Congruence can be proved using angle relationships such as alternate exterior, alternate interior, and corresponding angles. Use these as reasoning rather than statements. Transformation Congruence Justification (four statements): 1. Polygon A B C is obtained by a (reflection, translation, rotation) of polygon ABC. 2. A (reflection, translation, rotation) is a rigid motion. 3. Rigid motions preserve size and shape. 4. Therefore, polygon A B C is congruent to polygon ABC. Rigid Motions and Symmetry: Translations A rigid motion that shifts a linear object across a plane. The shift can be vertically or horizontally. When translating a shape, this preserves both tilt and orientation. Notation is written like this T <Δx, Δy>. Two translations are the same as one translation. Reflection A rigid motion that mirrors an image across the line of reflection. The reflection can be over any line. When reflecting a shape, tilt and orientation may or may not be preserved. Notation is written like this R k. K is the line of reflection; may be an axis or a custom line. Two reflections are the same as one reflection. Rotation A rigid motion that turns an image about a point. This rotation preserves orientation, but does not preserve tilt. Notation is written like this R <x, y> t. rotating about a point is always counterclockwise. When notation is expressed without coordinates, the rotation is about the origin. Every 90 degree rotation about the origin, the coordinates flip and the first coordinate is negated. This trick only works when rotating about the origin. Two rotations are the same as one rotation. Symmetry There are three main types of symmetry, identity, rotational, reflectional. Figures can have more than one type of symmetry and more than one symmetry of each type. Every figure has at least one symmetry which is its identity. Rotational Symmetry Divide the number of sides of a regular polygon by 360 degrees. Number of Symmetries of Regular Polygons: Triangle (3 sides): 3 reflectional, 2 rotational, 1 identity Pentagon (5 sides): 5 reflectional, 4 rotational, 1 identity Square (4 sides): 4 refletional, 3 rotational, 1 identity Hexagon (6 sides): 6 reflectional, 5 rotational, 1 identity