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Geometry, 2.1 Notes – Perpendicularity Parallel and perpendicular are opposite. Parallel = _______________________ Perpendicular = ______________________________ Perpendicular, right angles, 90° angles, all go together. Do not assume something is perpendicular on a diagram: For x-y plots, the x and y axes are perpendicular: Practical use of Geometry: constructing perpendicular line segments, bisecting line segments… Example problems: Given AB ⊥ CB Find m∠ABD Given: AB ⊥ BC , DC ⊥ BC Prove: ∠B ≅ ∠C Find the area of shape CABD. Geometry, 2.2 Notes – Complementary and Supplementary Angles Complementary angles are _______________________________________ (Each angle is called the ______________ of the other angle.) Supplementary angles are ________________________________________ (Each angle is called the ______________ of the other angle.) Are these pairs of angles complementary, supplementary or neither? Example: ∠A = 40° What is the complement of ∠A ? What is the supplement of ∠A ? Example: One of two supplementary angles is 3 times as big as the other. Find the measures of the two angles. Example: Given: Diagram as shown. Conclusion (prove): ∠1 is the supplement of ∠2 Geometry, 2.3 Notes – Making conclusions Procedure for making new conclusions… 1. Memorize, keep lists of theorems, definitions, and postulates (things you know). 2. Look for key words and symbols given in the problem. 3. Think of all the theorems, definitions and postulates (things you know) that mention those key words or symbols. 4. Decide which theorem, definition or postulate (thing you know) allows you to make a conclusion. 5. Make the conclusion and give reasons to justify the conclusion. Example: Given: AB bisects ∠CAD Conclusion: ? Example: Given: ∠A is a right angle. ∠B is a right angle. Conclusion: ? Example: Given: E is the midpoint of SG Conclusion: ? Example: Given: ∠PRS is a right angle. Conclusion: ? Some theorems, definitions and postulates you already know: (definition): An acute angle is an angle with measure between 0 and 90 degrees. (definition): A right angle is an angle whose measure is 90 degrees. (definition): An obtuse angle is an angle whose measure is between 90 and 180 degrees. (definition): A straight angle is an angle whose measure is 180 degrees. (definition): Congruent angles have the same angle measure. (definition): Congruent line segments have the same length. (definition): Points that lie on the same line are called collinear. (definition): A midpoint bisects a line segment into 2 congruent segments. (definition): An angle bisector bisects an angle into 2 congruent angles. (definition): A trisector divides a line segment or angle into 3 congruent pieces. (theorem): If 2 angles are right angles, then they are congruent. (theorem): If 2 angles are straight angles, then they are congruent. (theorem): If a conditional statement is true, then the contrapositive of the statement is true. (definition): Lines, rays, or line segments that intersect at right angles are perpendicular. (definition): Complementary angles are 2 angles that add to 90 degrees. (definition): Supplementary angles are 2 angles that add to 180 degrees. Geometry, 2.4 Notes – Congruent Supplements and Compliments Find ∠1 and ∠2 (theorem) If angles are supplementary to the same angle, then________________________________________ Find ∠1 and ∠2 (theorem) If angles are supplementary to ____________________, then__________________________________________ Find ∠1 and ∠2 (theorem) If angles are complementary to the same angle, then________________________________________ Find ∠1 and ∠2 (theorem) If angles are complementary to ___________________, then________________________________________ Example: Find ∠1 Example: Given: ∠A is complementary to ∠C ∠DBC is complementary to ∠C Conclusion: ? Geometry, 2.5 Notes – Addition and Subtraction Properties Addition Properties: If ________________________ are added to ____________________ the sums are _______________________ If ________________________ is added to____________________ the sums are ________________________ If _____________________ are added to____________________ the sums are ________________________ If _____________________ is added to____________________ the sums are ________________________ Example: In the diagram below, does AD = CB? Subtraction Properties: If ______________________________ are subtracted from ____________________________ the differences are _______________________ If ______________________________ is subtracted from _____________________________ the differences are ________________________ Example: ∠QUR ≅ ∠TUS , m∠QUS = 80° , m∠RUS = 20° Find m∠SUT Example: Is ∠GBI ≅ ∠QIB ? Geometry, 2.6 Notes – Multiplication and Division Properties Example: B, C trisect AD R, S trisect QT If AB=3 and QR=3, what can we say about AD and QT ? Example: KO bisects ∠JKM PS bisects ∠NPR If m∠JKO = 25° and m∠NPS = 25° What can we say about ∠JKM and ∠NPR Multiplication Property: If segments (or angles) are congruent, their like multiples are congruent. Division Property: If segments (or angles) are congruent, their like divisions are congruent. Example: ∠ ≅∠ EFG ABC BD and FH are bisectors m∠ABD = 30° m∠EFG = 4 x + 20 Find x Geometry, 2.7 Notes – Transitive and Substitution Properties Transitive Property: If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other. If angles (or segments) are congruent to congruent angles (or segments), they are congruent to each other. Example: Substitution Property: Example: If ∠A ≅ ∠B , find m∠A Example: If ∠1 is complementary to ∠2 and ∠2 ≅ ∠3 then: Geometry, 2.8 Notes – Opposite Rays and Vertical Angles Opposite Rays: ______________________________________________________ Examples: Vertical Angles: ______________________________________________________ Example: Vertical Angles are ________________________ Why? Examples: m∠AED = 60° Find m∠BEC m∠DHG = 120° ∠DHG ≅ ∠AGF Find m∠BGH Practice Problems: #1. Name 2 pairs of opposite rays: #2. m∠EFC = 100° , Find m∠AFD #3. m∠SOR = 70° , ∠PRN ≅ ∠SOR Find m∠MRO #4. m∠DBE = 40° , Find m∠ABC