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3 parallel and perpendicular lines annotated

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  Printed Name: Geometry Unit 3 Parallel and Perpendicular Lines Lesson Notes 3 1 Lines and Angles Objectives:    I will be able to identify relationships between figures in space.    I will be able to identify angles formed by two lines and a transversal. Do Now   Solve for each variable. Explain your reasoning. 4b + 43 = 6b + 17 b = 13 7a + 8 = 8a –   3 a = 11 Parallel lines  are coplanar lines that do not intersect. The symbol ∥   means “is  parallel to.”  ⃡ ∥ ⃡    ⃡ ∥ ⃡   Skew lines  are noncoplanar; they are not parallel and do not intersect.  ⃡  and ⃡  are skew. Parallel Planes  are planes that do not intersect. Plane  ABCD   ∥  plane  EFGH    Consider the figure below. a. Which three segments are parallel to ̅ ? ̅ , ̅ , ̅   b. Which four segments are skew to ̅ ? ̅ , ̅ , ̅ ,  ̅  c. What are two pairs of parallel planes? Name three pairs. VYXW ∥   RUTS, WXTS ∥ VYUR, VWRS ∥   YXTU   d. What are two segments parallel to plan  RUYV  ? ̅ , ̅   A transversal  is a line that intersects two or more coplanar lines at distinct  points. Angles 3 , 4 , 5 , and 6  lie between l  and m . They are interior   angles. Angles 1 , 2 , 7 , and 8  lie outside of l  and m . They are exterior   angles.    Transversal : a line that intersects two or more coplanar lines at different points. The angles formed by two lines and a transversal are given special names. Corresponding Angle: two a ngles that occupy corresponding  positions. ∠   1  and ∠   5  are corresponding angles.  Alternate Exterior Angle:  two angles that lie outside the two lines on opposite sides of the transversal. ∠   2 and ∠   8 are alternate exterior angles. List all pairs of angles that fit the description.  Alternate Interior Angles:  two angles that lie between the two lines on opposite sides of the transversal. ∠   3  and ∠ 5  are alternate interior angles. corresponding alternate exterior Alternate interior Consecutive interior ∠  1 and ∠  5 ∠  2 and ∠  8   ∠  3and ∠  5   ∠  4 and ∠  5   ∠  4 and ∠  8   ∠  1 and ∠  7   ∠  4 and ∠  6   ∠  3 and ∠  6   ∠  2 and ∠  6   ∠  3 and ∠  7   Same-Side Interior Angles:   two angles that lie between the two lines on the same side of the transversal. ∠   4  and ∠   5  are same-side interior angles. (aka consecutive interior angles) Assignment:  pp. 144-146; 12-22E, 25-28, 48.48.  3 2 Properties of Parallel Lines Objectives:    I will be able to prove theorems about parallel lines.    I will be able to use properties of parallel lines to find angle measures. Do Now 1. CE and AC are ______perpendicular _______. 2. EH and AB are _______skew _____________. 3. Plane DBG and plane AHE are _parallel _____. Use the figure below. Fill in the blank with parallel, skew, or perpendicular. Postulate 3-1.  Same-Side Interior  Angles Postulate . If a transversal intersect two parallel lines, then same-side interior angles are supplementary. If l   ∥   m , then m ∠  4 + m ∠  5 = 180 m ∠  3 + m ∠  6 = 180 Theorem 3-1.   Alternate Interior  Angles Theorem.  If a transversal intersects two parallel lines, then alternate interior angles are congruent. If l   ∥   m , then ∠  4  ≅   ∠  6  ∠  3 ≅   ∠  5  Theorem 3-2.  Corresponding  Angles Theorem.  If a transversal intersects two parallel lines, then corresponding angles are congruent. If l   ∥   m , then ∠  1  ≅   ∠  5  ∠  2 ≅   ∠  6 ∠  3 ≅   ∠  7  ∠  4 ≅   ∠  8  Theorem 3-3.   Alternate Exterior  Angles Theorem.  If a transversal intersects two parallel lines, then alternate exterior angles are congruent. If l   ∥   m , then ∠  1  ≅   ∠  7  ∠  2 ≅   ∠  8  Given:   a   ∥   b   Statements Reasons 1. a ∥  b 1. Given  2. ∠ 1 ≅ ∠ 5   2. Corresponding angles are congruent.  3. ∠ 7 ≅ ∠ 5   3. Vertical angles are congruent.  4. ∠ 1 ≅ ∠ 7   4. Transitive Property of Congruence.  Prove: ∠  1 ≅   ∠  7      Given:   a   ∥   b   Statements Reasons 1. a ∥  b 1. Given  2. ∠ 1 ≅ ∠ 2   2. Alternate exterior angles are congruent  3. ∠ 2 & ∠ 3 are sup.   3. Linear Pairs are supplementary.  4. ∠ 1 & ∠ 3 are sup.   4. Congruent Supplements Theorem  Prove: ∠  1 and   ∠  3 are supplementary  Application. Use properties of parallel lines to find the value of x. a. Alternate Exterior Angles Justify your answers by referring to specific Theorems. x - 8 = 55 x = 63 Linear Pairs  b. Corresponding Angles (14x + 7) + 103 = 180 14x + 110 = 180 14x = 70 x = 5 c. Linear Pairs Corresponding Angles (4x  –   3) + 135 = 180 4x + 132 = 180 4x = 48 x = 12 d. Alternate Exterior Angles 9x + 7 = 115 9x = 108 x = 12 Assignment:  pp. 8-20E, 26.
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