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Transcript
2.5 Angle Relationships ________________________ - Two adjacent angles whose distinct sides lie on the same line. ________________________ - Two nonadjacent angles formed by two intersecting lines. __________________________ - A statement that can be expressed in “if-then” form. IF a, THEN b. If an object is a triangle, then it is a polygon. If a number is even, then it is divisible by two. ________________________________________ - The statement formed by interchanging the hypothesis (the “if” part) and conclusion (the “then” part) of a conditional statement. IF b, THEN a. If an object is a polygon, then it is a triangle. If a number is divisible by two, then it is even. _____________________________________ - The statement formed by negating both the hypothesis (the “if” part) and the conclusion (the “then” part) of a conditional statement. IF NOT a, THEN NOT b. If an object is not a triangle, then it is not a polygon. If a number is not even, then it is not divisible by two. ___________________________________ - The statement formed by interchanging and negating both the hypothesis (the “if” part) and the conclusion (the “then” part) of a conditional statement. (The inverse of the converse of a conditional statement.) IF NOT b, THEN NOT a. If an object is not a polygon, then it is not a triangle. If a number is not divisible by two, then it is not even. Geometry Lesson 2.5: Angle Relationships Page 1 Investigation: Linear Pair Conjecture Step 1 Draw ⃡𝑃𝑄 and place a point R between P and Q. Choose another point S not on ⃡𝑃𝑄 and draw 𝑅𝑆. You have just created a linear pair of angles. Place the “zero edge” of your protractor along ⃡𝑃𝑄 . What do you notice about the sum of the measures of the pair of angles? Step 2 Compare. Does everyone make the same observation? Complete the statement. If two angles form a _____________, then the measures of the angles add up to _____. D mACD + mDCB = ______ A B C Investigation: Vertical Angles Conjecture Step 1 Draw two intersecting lines onto patty paper. Label the angles as shown in the box below. Which angles are vertical angles? Step 2: Fold the paper so that the vertical angles lie over each other. What do you notice about their measures? Step 3 Repeat this investigation with another pair of intersecting lines. Step 4 Compare your results with the results of others. Complete the statement. If two angles are __________________, then they are _____________. 1 4 2 1 3 and 2 4 3 Geometry Lesson 2.5: Angle Relationships Page 2 Example 1: Write the converse, inverse, and contrapositive of each and tell whether each is true or false. a) Linear pair conjecture: converse: inverse: contrapositive: b) Vertical angle conjecture: converse: inverse: contrapositive: Example 2: Find x, y, and z. 135 z x y Geometry Lesson 2.5: Angle Relationships Page 3 Example 3: Find all the missing angles. 109 a c e d 50 b 150 a = ______ b = ______ d = _____ c = ______ h e = ______ f = ______ 130 g f 55 k i j 120 v l q m 110 n 105 145 p u r t o s g = ______ h = ______ i = ______ p = ______ q = ______ r = ______ j = ______ k = ______ l = ______ s = ______ t = ______ u = ______ m = ______ n = ______ o = ______ v = ______ Geometry Lesson 2.5: Angle Relationships Page 4 Example 4: Find the value of x in each. 20x - 40 5x + 20 6x + 10 11x - 15 22x - 34 5x + 25 15x - 30 8x - 20 4x + 20 7x + 50 Homework: pp. 122 – 125 => 1 – 10; 13 – 17; 19; 22 – 27 Geometry Lesson 2.5: Angle Relationships Page 5