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GEOMETRY – CHAPTER 3 SUMMARY A. Vocabulary Term Parallel Lines (pg 24) Definition & symbols Picture and/or example Transversal (pg 127) Corresponding Angles (pg 127) Alternate Interior Angles (pg 127) Same-side Interior Angles (Consecutive Angles) (pg 127) Alternate Exterior Angles Same-side Exterior Angles Geometry – Ch. 3 Summary pg 1 of 11 Flow Proof (pg 135) Equiangular triangle (pg 148) Acute Triangle (pg 148) Right triangle (pg 148) Obtuse triangle (pg 148) Equilateral triangle (pg 148) Isosceles Triangle (pg 148) Geometry – Ch. 3 Summary pg 2 of 11 Scalene Triangle (pg 148) Exterior angle of a polygon (pg 149) Remote Interior Angles (pg 149) Polygon (pg 157) Convex polygon (pg 158) Concave polygon (pg 158) Geometry – Ch. 3 Summary pg 3 of 11 Regular Polygon (pg 160) Slope (pg 165) Slope-intercept form of a linear equation (pg 166) Standard form of a linear equation (pg 167) Point-slope form of a linear equation (pg 168) Perpendicular Lines (pg 45) Geometry – Ch. 3 Summary pg 4 of 11 B. Postulates/Theorems Name Description Picture and/or example Postulate 3-1 & Corresponding Angles Postulate (pg 128) Theorem 3-1: Alternate Interior Angles Theorem (pg 128) Theorem 3-2: Same-Side (Consecutive) Interior Angles Theorem (pg 128) Theorem 3-3: Alternate Exterior Angles Theorem (pg 130) Theorem 3-2: Same-Side (Consecutive) Exterior Angles Theorem (pg 130) Geometry – Ch. 3 Summary pg 5 of 11 Postulate 3-2 & Theorems 3-5 through 3-8 (These are converses of Postulate 3-1 & Theorems 3-1 through 3-4) If two lines and a transversal form • corresponding angles that are ____________________ • alternate interior angles that are ________________ or • same-side (consecutive) interior angles that are ___________________ or • alternate exterior angles that are _____________________ • same-side (consecutive) exterior angles that are ___________________ or then the lines are __________________ Theorem 3-9 (pg 141) Theorem 3-10 (pg 141) Theorem 3-11 (pg 142) Theorem 3-12 Triangle Sum Theorem (pg 147) Geometry – Ch. 3 Summary pg 6 of 11 Theorem 3-13 Triangle Exterior Angle Theorem (pg 149) Theorem 3-14 Polygon AngleSum Theorem (pg 159) Theorem 3-15 Polygon Exterior Angle-Sum Theorem (pg 160) Slopes of Parallel Lines Slopes of Perpendicular Lines Geometry – Ch. 3 Summary -20 -20 -15 pg 7 of 11 -15 -10 -5 -10 -5 D. Practice writing proofs: 1. Fill in the blank to complete the two-column proof. Given : j // k. Prove: ∠ 2 ≅ ∠ 3 (You are proving the Alternate Interior Angles Theorem here, so do not use it as a reason.) € €€ Statements 1 2 3 k Reasons j // k ∠1 ≅ ∠3 ∠1 ≅ ∠2 €€ €€ ∠2 ≅ ∠3 2. Rewrite the above proof as a flow proof. €€ 3. Given: j // k, ∠1 ≅ ∠2 Prove: m // p m p 1 j € € € 2 k 3 Geometry – Ch. 3 Summary j pg 8 of 11 E. Equations of Lines: 1. Arrange the following lines in order based on the values of their slopes, from smallest to greatest. Explain. Line k Line l Line m Line n Line p 2. Estimate the value of the slope for each line. a) b) 3. Find the value of the slope for the line that passes through each pair of points. Use both formula and graph. a) (2, -3) and (0, 4) b) (1, 5) and (1, -2) 12 10 8 6 4 2 4. Explain how to graph a line given its equation in slope-intercept form y = mx + b. -20 Examples: Graph y = 3x – 2 -15 -10 -20 Graph y = 3. 2 3 Graph y = − x + 1 € -5 -15 -10 Graph x = - 4. 12 12 12 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 5. Describe step-by-step how to write the equation of a line in slope-intercept form if you have: * A point (x1, y1) and the slope m. Ex: Write the equation of a line with slope m = 5 and passing through the point (1, 0). -20 -15 -10 Geometry – Ch. 3 Summary -20 -5 -15 -10 -20 -5 -15 -10 -20 pg 9 of 11 -5 -15 -10 -5 * Two points (x1, y1) and (x2, y2). Ex: Write the equation of a line passing through the points (3, -1) and (-2, 5) a) using the point-slope form. a) using the slope-intercept form. * Another line parallel to it and a point on it. Ex: Write the equation of a line that is parallel to the line y = 2x + 5 and contains the point (1, -3). Graph both lines. * Another line perpendicular to it and a point on it. Ex: Write the equation of a line that is perpendicular to the line y = 2x + 5 and contains the point (1, -3). Geometry – Ch. 3 Summary -20 -15 -10 -20 -15 -10 pg 10 of 11 F. Constructions: Using only a compass and a straightedge, construct the following. 1. A line parallel to a given line and through a given point that is not on the line. (pg 181) A j Explain why this construction is valid. 2. A line perpendicular to a given line and through a given point on the line. (pg 182) 3. A line perpendicular to a given line and through a given point that is not on the line. (pg 183) A j j Geometry – Ch. 3 Summary pg 11 of 11