Unit 1 Thm 1 - if 2 angles are right angles, then they are congruent. Thm 2 - If 2 angles are straight angles, then they are congruent. Defn - If an angle is a 90 degree angle, then it is a right angle. Defn - If an angle is a 180 degree angle, then it is a straight angle. Defn - If a ray bisects an angle, then it divides the angle into 2 congruent angles. Defn - If a line bisects a segment, then it divides the segment into 2 equal segments. Defn - If a point divides a segment into 2 equal segments, then it is the midpoint of a segment. Thm 3- If a conditional statement is true, then the contrapositive is also true (if p then q, if ~p, then ~q). Unit 2 Thm 4- If angles are supplementary to the same angle, then they are congruent. Thm 5- If angles are supplementary to congruent angles, then they are congruent. Thm 6- If angles are complementary to the same angle, then they are congruent. Thm 7- If angles are complementary to congruent angles, then they are congruent. Thm 8/9- If a segment (or angle) is added to two congruent segments (or angles), the sums are congruent.(Addition property) Thm 10/11- If congruent segments (or angles) are added to congruent segments (or angles), the sums are congruent.(Addition property) Thm 12- If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) Thm 13- If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) Thm 14: If segments (or angles) are congruent, then their like multiples are congruent. (Multiplication Property) Thm 15: If segments (or angles) are congruent, then their like divisions are congruent (Division Property) Thm 16: If angles (or segments) are congruent to the same angle (or segments), then they are congruent. (Transitive Property) Thm 17: If segments (or angles) are congruent to congruent angles (or segments), then they are congruent (Transitive Property) Thm 18: Vertical angles are congruent Unit 3 Congruent figures: all corresponding sides and angles are congruent Reflexive Property: any segment or angle is congruent to itself SSS Postulate: if all 3 sides of one triangle are congruent to all 3 sides of another triangle, then the triangles are congruent SAS Postulate: If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent ASA Postulate: If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent. Thm 19: all radii of a circle are congruent CPCTC: Corresponding Parts of Congruent Triangles are Congruent Median: a line segment drawn from any vertex to the midpoint of the opposite side Altitude: a line segment drawn from any vertex that is perpendicular to the opposite side Auxiliary line: two points determine a line (this is a line that is added to a diagram) Thm 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent Thm 21: If two angles of a triangle are congruent, the sides opposite the angles are congruent Thm 20 Inverse: If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. Thm 21 Inverse: If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle. HL Postulate: If you have two right triangles where the hypotenuse and leg of one triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent. Reasons: ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● If two angles form a straight angle, then they are supplementary. (Angle SVY is supp. To angle TVA) If a segment from a vertex of a triangle divides the opposite side into 2 congruent segments, it is a median. (SZ is the median of TA) (FD is the median) If a ray divides an angle into 2 congruent angles, it bisects it/the angle. (MJ bisects angle AMP) Two points determine a seg ( Auxiliary line ) A bis divides an angle into 2 congruent angles. (NRO is congruent to ORA - given statement is that NR is the bisector of angle ORA) Supp angles are congruent. An alt. of a triangle forms right angles with the side to witch it is drawn ( Angle FDE and AFM are right angles) A mid. Divides a seg into 2 congruent segs. (When mid is given, two segs are congruent) If 2 angles form a straight angle, they are supplementary. (Angle 2 is sup to angle EFG) If two lines intersect to form a right angle, they are perpendicular (PR is perpendicular to RS) If two angles are perpendicular, then they form a right angle. Supplements of congruent angles are congruent. (1 is congruent to 2) Complements of congruent angles are congruent. Compliments of the same angle are congruent. Supplements of the same angle are congruent. If two angles are right angles, then they are congruent. ● ● ● ● ● ● ● ● ● Altitude forms right angles (Angle BAC and CNG are right) If a seg is perpendicular to the opposite side in a triangle, then it is the altitude (AD is the altitude) If two angles are both supp. and congruent, then they are right angles. (used to prove an altitude/rights angles) An alt. Of a triangle is perpendicular to one of the sides. (BD is an alt) If a seg frm a vertex of a triangle is perpendicular to the opposite side, it is an altitude of the triangle. (AD is an alt) A median of a triangle divides one side into 2 congruent segs (AD is congruent to DC --- given that BD is a median) If a point divides a seg into two congruent segs, the point is a midpoint. (M is midpoint of BC) If a point is a midpt. Of a segment, it divides the seg into 2 congruent segs. (FG is congruent to HG - given that G is the mdpt. Of FH) Corr angles congruent → parallel lines (coressponding) Bisector: 2 congruent angles Midpoint: 2 congruent segments Midean: same as before. Trisect: 3 congruent segs. Altitude: perpendicular (90 degree) angles. Unit 4 Thm 23: If two angles are both supplementary and congruent, then they are right angles. Thm 24: If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment. Thm 25: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment. Transversal: a line that intersects two coplanar lines in two distinct points Alternate Interior angles:a pair of angles formed by two lines and a transversal. The angles must be inside the figure and on alternate sides of the transversal. Alternate Exterior angles:a pair of angles formed by two lines and a transversal. The angles must be outside the figure and on alternate sides of the transversal. Corresponding angles: a pair of angles formed by two lines and a transversal. One angle must lie inside the figure and the other outside the figure. They both need to be on the same side of the transversal. Parallel Lines: two lines that do not intersect Slope: the change in y over the change in x Thm 26: If two nonvertical lines are parallel, then their slopes are equal. Thm 27: If the slopes of two nonvertical lines are equal, then the lines are parallel. Thm 28: If two lines are perpendicular and neither is vertical, each line’s slope is the opposite reciprocal of the other’s. Thm 29: If a line's slope is the opposite reciprocal of another line’s slope, the two lines are perpendicular. Unit 5 Thm 31: Alternate interior angles congruent ---> parallel lines (and the converse is true) Thm 32: Alternate exterior angles congruent ---> parallel lines (and the converse is true) Thm 33: Corresponding angles congruent ----> parallel lines(and the converse is true) Thm 34: Same side interior angles supplementary ---> parallel lines (converse is true) Thm 36: If 2 coplanar lines are perpendicular to a third lines, they are parallel Thm 43: In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other. Thm 44: If two lines are parallel to a third line, they are parallel to each other. (transitive property of parallel lines) Quadrilateral Properties Document - - If a line's slope is the opposite reciprocal of another line’s slope, the two lines are perpendicular. If the slope is congruent, then they are parallel. Not a postulate or anything but if we have parallel lines, all angle pairs are either congruent or supplementary. So it’s either a parallelogram, rectangle, rhombus or square. 1. Check to see if opposite sides are parallel (If yes, then it’s a parallelogram) 2. Check to see if consecutive sides are perpendicular (If yes, then it’s a rectangle. If no, then try step 3 to see if it’s a rhombus) 3. Check to see if diagonals are perpendicular (If all 3 steps are a yes, then it’s a square. If step 2 was no, but this is yes, then it’s a rhombus because it’s the only parallelogram that has perpendicular diagonals without perpendicular sides.. If only step 1 is yes, then it’s a parallelogram) Unit 6 Foot: the point of intersection of a line and a plane is called the foot of the line. 4 ways 1. 2. 3. 4. to determine a plane: Three noncollinear points determine a plane A line and a point not on the line determine a plane. Two intersecting lines determine a plane. Two parallel lines determine a plane. Postulate: If a line intersects a plane not containing it, then the intersection is exactly one point. Postulate: If two planes intersect, their intersection is exactly one line. Definition: A line is perpendicular to a plane if it is perpendicular to every one of the lines in the plane that pass through its foot. Thm 48: If a line is perpendicular to two distinct lines that lie in a plane and that pass through its food, then it is perpendicular to the plane. ● Two parallel lines deterinme a plane Unit 7 Thm 50: The sum of the measures of the three angles of a triangle is 180. Thm 51: The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Thm 52: A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one half the length of the third side. (Midline Theorem) Thm 53: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. (No Choice Theorem) Thm 54: AAS can be used to prove two triangles congruent. Thm 55: The sum of the measures of the angles of a polygon with n sides if given by the formula S = 180(n-2) Thm 56: The sum of the exterior angles in a polygon is always 360 degrees. Thm 57: The number of diagonals that can be drawn in a polygon of n sides is given by the formula d = n(n-3)/2 A regular polygon has equal side lengths and equal angle measurements. Thm 58: The measure E of each exterior angle of an equiangular polygon of n sides if E = 360/n Unit 8 A Ratio is a quotient of two numbers A Proportion is an equation stating that two or more ratios are equal Thm 59: In a proportion, the product of the means is equal to the product of the extremes (Means-Extremes Product Theorem) Thm 60: If the product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then either pair of numbers may be made the extremes and the other pair the means, of a proportion (Means Extremes Ratio Theorem) Similar Polygons have corresponding sides congruent and corresponding angles congruent Thm 61: The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides Thm 62: If two angles in one triangle are congruent to two angles in another triangle then the two triangles are similar. (AA similarity) Thm 63: If the ratios of corresponding sides on two triangles are congruent, then the triangles are similar.(SSS similarity) Thm 64: If the ratios of two corresponding sides in two triangles are congruent AND the included angles are congruent, then the triangles are similar (SAS similarity) Thm 65: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally. (Side Splitter Theorem) Thm 66: If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally Thm 67: If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides (Angle Bisector Theorem)
