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1st Semester Geometry Notes page 1 1-3 Points, Lines, Planes A point A Points A, B are collinear Points A, B, and C are coplanar plane M or plane ABC (name with 3 pts) •Intersection of two distinct lines is a point •Intersection of two distinct planes is a line 1-4 Segments, Rays, Parallel Lines, Planes Segment AB or •Opposite rays share same endpoint •Opposite rays are collinear Parallel Lines -Never intersect -Extend in the same directions -Coplanar Skew Lines -Never intersect -Extend in different directions -Noncoplanar Parallel Planes -Can never intersect 1-5 Measuring Segments AB is the abbreviation for the distance between points A and B. Segment Addition Postulate Midpoint •B is exactly halfway between A and C •B is the average coordinate of A and C 1-6 Measuring Angles vertex Congruent Angles m 1 = m 2 (the measure of angle 1 equals the measure of angle 2) 1 ≅ 2 (Angle 1 is congruent to angle 2) (May also be indicated by arc on both angles) 1st Semester Geometry Notes page 2 Pairs of Angles 1 and 2 are adjacent angles -No interior points in common -Share the same vertex R -Share common side Vertical Angles -Non-adjacent; -Formed by two intersecting lines -Are congruent 1 and 3 Angle Addition Postulate mAOB + mBOC = mAOC 2 and 4 Linear Pairs -Form a straight angle -Are supplementary (sum = 180) Compl. Corner Suppl. Straight 1-1 Inductive Reasoning Law of Detachment: If a, then b. (True) Given: a is True b is therefore True. Law of Syllogism If a, then b. (True) If b, then c. (True) If a, then c must be True. If A is falls to the right, then B falls to the right A B Given: A falls to the right is True Then: B falls to the right. 2-1 2-2 2-3 5-4 Deductive Reasoning Conditional: If a, then b statement a is the hypothesis; b is the conclusion Converse: Switch the hypothesis and conclusion. If b, then a. Truth value of a statement: Either True or False, where True means always true Biconditional: Both the conditional and its converse are true. You can combine both statements with if and only if. a if and only if b. 2-4 Algebraic Properties A B C If B is falls to the right, then C falls to the right. If A falls to the right, then C falls to the right. Addition Property If a = b, then a + c = b + c Subtraction Property If a = b, then a – c = b – c Multiplication Property If a = b, then a * c = b * c Division Property If a = b and c ≠ 0, then a/c = b/c Reflexive Property a=a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c Substitution Property If a = b, then b can replace a in any expression Distributive Property a(b + c) = ab + ac 1st Semester Geometry Notes page 3 1-8 8-1 Pythagorean Theorem, Midpoint, Distance Formula Pythagorean Theorem Classifying Triangles (leg1)2 + (leg2)2 = hypotenuse2 True only for right triangles Distance between 2 points -Use the Pythag. Thm Let a, b, c be the lengths of the sides of a triangle, where c is the longest Acute: c2 < a2 + b2 Right: c2 = a2 + b2 Obtuse: c2 > a2 + b2 Midpoint (average x, average y) 3-1 3-2 3-3 Parallel Lines and Angles Transversal: line that cuts across two or more lines Congruent if and only if l and m are parallel Vertical angles are congruent Linear pairs are supplementary 3-4 Triangle Sum Thm, Exterior Angle Thm 4-5 Isosceles and Equilateral Triangles A triangle is isosceles if and only if the base angles are congruent. A triangle is equilateral if and only if the triangle is equiangular Same-side exterior: ∠1 and ∠4 ∠5 and ∠8 Supplementary if and only if l and m are parallel 3-5 Polygon Angle Sum Thms n = number of sides 1st Semester Geometry Notes page 4 Interior angle Exterior angle 360 for regular polygons 180 – n 3-6 3-7 Graphing Equations of Lines •Any point on the line must satisfy the equation of the line (y = mx + b) •Parallel lines have equal slopes (same steepness) •Perpendicular lines have slopes that are negative reciprocals of each other Standard Form: Ax + By = C Point-Slope Form: y – y1 = m (x – x1) 9-2 Reflections: Preimage and image are -on opposite sides of line of reflect. -equidistant from line of reflection Reflect about x-axis (x, y) → (x, -y) Reflect about y-axis (x, y) → (-x, y) Reflect about y = x (x, y) → (y, x) 9-1 Translations: (x, y) → (x+a, y+b) 9-3 Rotations about origin: For each 90° of rotation, switch the x and y coordinates; then determine signs based on the quadrant after rotation Preimage: before the transformation Image: after the transformation Isometry: size and shape stay the same Reflections, Translations, and Rotations are isometries 9-5 Dilations 9-4 Symmetry Enlargement: Multiply both x and y by a scale factor k greater than 1 (x, y) → (kx, ky) Reduction: Multiply both x and y by a scale factor k between 0 and 1 (x, y) → (kx, ky) Vertical stretch Multiply the y only by a scale factor k greater than 1 (x, y) → (x, ky) Horizontal shrinkage Multiply the x only by a scale factor k between 0 and 1 (x, y) → (kx, y) 1st Semester Geometry Notes page 5 7-1 Ratios and Proportions 7-2 Similarity 2 4 7-3 Proving Triangles Similarity SSS, SAS, AA = 3 6 1st Semester Geometry Notes page 6 7-4 Similarity in Right Triangles 1) 2) 3) Redraw and label triangles. Fill in the table with given information Use proportions or Pythag. Thm to solve for missing lengths Small leg Medium leg Hypotenuse ∆1 a b c ∆2 r h a ∆3 h s b 7-5 Proportions in Triangles in a triangle 4-1 Congruent Polygons Two polygons are congruent if they have the same size and shape. Two polygons are congruent if and only if all corresponding sides and corresponding angles are congruent. 1st Semester Geometry Notes page 7 4-2 4-3 4-6 Proving Triangles Congruent SSS SAS ASA AAS HL 4-4 4-7 Using CPCTC in Proofs CPCTC is an abbreviation of the phrase “Corresponding Parts of Congruent Triangles are Congruent.” 1st Semester Geometry Notes page 8 5-5 Triangle Inequalities Given two sides a and b, the third side the triangle with length c must satisfy |a–b|<c<a+b 5-1 5-2 5-3 Special Segments in Triangles Altitudes (vertex to opposite side at right angles – Orthocenter Perpendicular Bisectors (90 degrees through midpoint of a side) – Circumcenter Medians (vertex to midpoint of opposite side) – Centroid Angle Bisectors (vertex to opposite side through line splitting the vertex angle in half– Incenter