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3.1 Angles and Their Properties _________________ -- the union of 2 rays that have the same endpoint (symbol: ) What is the name of this angle? ________________ Which point is the vertex of this angle? ________________ Name the sides of this angle. _______ & _______ _________________ -- is another name for a line. Draw an example below! _________________ -- is another name for a ray. Draw an example below! Give three different names to the angle at right. Why is the “A” in middle of each name? Draw a bisector to the angle at left. How many angles are there in the drawing now? _______. An ____________ _____________ must always divide an angle _____ ____________. Draw an angle and label the Interior and the Exterior. The exterior of an angle is always _________________________. When mentioning the angle , we write _________________. When mentioning the _______________ of an angle, we write __________________. Is o the angle bisector to XWZ? , ∠XWY = 79o, and m∠YWZ = 38o Find m∠ZWV. In the picture below, m∠YWZ = 5x – 11 and m∠YWX = 2x + 25. What is the value of x? What is the measure of each angle? What CAN you assume about the picture below? What CAN’T you assume about the picture below? * * * * * * 3.2 Arcs and Rotations Identify a minor arc. Identify a major arc. Name the circle. Name a central angle. The measure of a minor arc is ____________________ to the measure of the ____________________________ Using the circle at the top of the page, if measure of is 300O, what is the measure of ∠ABC?______________ What is the There are ____________ degrees in a circle. Draw a circle with a straight angle cutting through the center of the circle. This line ___________________ the circle. Each half of the picture you drew is called a ________________________________________. Draw four concentric circles below. The amount of degrees an object rotates is called its _________________________________. A rotation can be turned either ______________________________________ or ____________________________________________________ A positive magnitude is turned ___________________________________________________ A negative magnitude is turned __________________________________________________ Examples 3-2 1. m RWD = 113. m 2. Then what is m 3. A counterclockwise rotation (turn) has a _______________________ magnitude. 4. m RWD = 121. m 6. A rotation of -170° is the same as a rotation of _________________°. 7. When the hour hand of a clock moves from 3 to 6, what is the magnitude of the rotation? = ________ ? _____________ = ________ 3.3 Properties of Angles Name of Angle: Zero Angle Acute Angle Right Angle Obtuse Angle Straight Angle Complementary Angles Supplementary Angles Adjacent Angles Linear Pair Vertical Angles Picture: Symbols: Examples 3-3 1. Complementary angles ___________________________ have equal measures. (always, sometimes, never) 2. Two acute angles are ______________________ supplementary. (always, sometimes, never) 3. Vertical angles _______________________ have equal measure. (always, sometimes, never) 4. Supplementary angles are ___________________________ linear pairs. (Always, sometimes, never) 5. Right angles are __________________________ complementary. (Always, sometimes, never) 6. m 2 = 3x –6 m 3 = 17 – x x = __________ 7. m 2 = 3x – 6 m 4 = 7x + 8 m 2 = _______ 8. Two angles are supplementary. One measures 2x + 7, the other measures 3x – 2. Find the measure of each angle. 9. A support wire makes a 57° angle with the ground. What is the measure of the other angle formed by the wire and the support? 3.4 Properties For any real numbers a, b, c: **Reflexive Property of Equality: ____________________________ Example: **Symmetric Property of Equality:_____________________________ Example: **Transitive Property of Equality: _____________________________ Example: Which is which?? 1) If 1 2, and 2 3 , then 1 3 ________________________ 2) A A ___________________ 3) If N K , then K N __________________ **Addition Property of Equality: _______________________________ Example: **Multiplication Property of Equality:____________________________ Example: **Equation to Inequality: If a + b = c; then a< c and b < c Example: **Substitution Property: If a = b, then a may be substituted for b Example: 3.5 One Step Proofs A proof is a sequence of __________________________________ ________________________________________ that start with the _________________________________________ and end with the __________________________________________. A proof must always have _________________ information, and we are always trying to justify statements to come to a logical conclusion. Example: Given: 4r – 3 = 11 Prove: r = 3.5 Statement Reason What is a midpoint? _________________________________________________________ Draw line segment AB. Put C where you think the midpoint of the segment should be. Use "tick marks" to show equality. Given C is a midpoint above, what do we know? _____________________________________ Why do we know this is true? ____________________________________________________ Draw two lines that intersect and identify the four angles as angles 1, 2, 3,& 4 in a clockwise manner. Why is the measure of angle 1 = to the measure of angle 3? __________________________________________. Examples 3-5 State the justification in the following 1-step proof: 1. Conclusion Justification 0) Wis the midpoint of CR 1) CW = CR 2. Conclusion 1) Justification 0) ∠7 is a right angle 0) Given m∠7 = 90O 1) 3. Conclusion 1) Conclusion 1) Justification 0) ∠3 and ∠4 are complementary 4. 0) Given 0) Given m∠ 3 + m∠4 = 90 1) Justification 0) m∠5 = 35O 0) Given 1) ∠5 is acute 1) 5. State a conclusion and give its justification: 6. State a conclusion and give its justification: 7. State a conclusion and give its justification: 8. What kinds of statements can be used as justifications in proofs? 3.6 Parallel Lines What is a Transversal? _________________________________________________________. Which one is the transversal in the picture? How many angles were formed above? __________ Label the corresponding angles in the drawing above. Identify all pairs of corresponding angles in the picture: ___________________________________________________________________________ **If lines are parallel then all corresponding angles are _________________________. **If we know all corresponding angles are congruent, then we know what must be true about the lines that create them? _________________________________________________________. The two marked lines above are the _______________________________________________________ The formula to find the slope of two ordered pairs is: _________________________________. A horizontal line has a slope of ___________________________. A vertical line has a slope of _____________________________. All parallel lines slopes are _____________________________________. Examples: 1. What is the slope of the line through points (-2, 6) and (8, 11)? 2. What is the slope of the line through points (-23, -27) and (16, -46)? 3. If a line has an equation of 2x + 3y = 12, what is the slope of the line? (Solve for y) 4. If line one is parallel to line two, and line one has a slope of 2/3 what is the slope of line two? 5. If two lines are parallel to the same line, then the two lines must be ___________________ to each other. 6. Corresponding angles are _______________________ equal in measure. (Always, sometimes, never) 7. a // b. m∠3 = 134. m∠5 = ___________ 8. a // b. Which angle(s) have the same measure as angle 6? 3.7 Perpendicular Lines Define Perpendicular Lines: Draw two lines that are perpendicular and identify the four angles as 1, 2,3 & 4 in a clockwise manner. True or False: Angles 1 & 2 are complimentary. True or False: Angles 1 & 2 are supplementary. True or False: Angles 1 & 2 form a linear pair. Draw two lines, a & b, that are perpendicular to a third line, c. Put in the appropriate markings. What is true about lines a & b? Why is this true? If lines a & b from the previous drawing are parallel what is true about their slopes? Lines that are perpendicular have slopes that are the _________________ ___________________ to each other. If line x has a slope of 2/3 and line y is perpendicular to line x, then the slope of line y is? __________. To be perpendicular lines, the slopes of the two lines must multiply to give you _________________ Examples: 1. What is the slope of a line perpendicular to y = - 2. What is the slope of a line parallel to the line with the equation 5x + 4y = 12 3. What is the slope of a line perpendicular to 3x – 2y = 6? 4. If two lines are perpendicular, then their slopes would multiply to give you what number? ?