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Angle Relationships in Geometry

Explore the concepts of congruent, complementary, supplementary, adjacent, and vertical angles. Discover theorems and the properties of parallel lines.

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Angle Relationships in Geometry

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  1. Angle Relationships

  2. Terms 40o 40o CONGRUENT ANGLES: Two angles are congruent angles if and only if they have the same measure.

  3. Terms B D Side Side Vertex C A • A Pair of Complementary Angles – 2 angles whose measures have the sum of 90o • A Pair of Supplementary Angles – 2 angles whose measures have the sum of 180o • ADJACENT ANGLES: Angles that share a vertex and a side and whose interiors do not overlap Are complementary (/supplementary angles) adjacent angles?

  4. Terms D A P B C • A Pair of Vertical Angles – angles formed by 2 intersecting lines; they share a common vertex but not a common side – If AB and CD intersect at point P so that point P is between points A and B and also between points C and D, then APC and BPD are a pair of vertical angles. APD and BPC are also a pair of vertical angles.

  5. Terms W X Y Z • A Linear Pair of Angles -– adjacent angles whose noncommon sides are opposite rays -– If X, Y, Z are consecutive collinear points and W is a point not on XZ, then XYW and WYZ form a linear pair of angles. • Are supplementary angles a linear pair of angles? • How many linear pair of angles are formed when 2 lines intersect?

  6. Theorems • Linear Pair Thm - If two angles form a linear pair, then they are supplementary. • Vertical Angles Thm - If two angles are vertical angles, then they are congruent.

  7. a=127 a=c=68; b=112 a=c=35; b=40; d=70 a=c=20; b=d=70; e=110 a=70; b=55; c=25 a=b=90; c=42; d=48; e=132

  8. Special Angles formed by a Transversal

  9. Perpendicular Lines () – two lines that intersect to form a right angle Parallel Lines (//)– 2 or more lines that are coplanar and that do not intersect Skew Lines – lines that are not coplanar and that do not intersect Why is ‘coplanar’ not in the definition of  lines? Parallel Lines Perpendicular Lines

  10. TRANSVERSAL - A line that intersects 2 or more coplanar lines at different points l2 l3 l3 l2 l3 l2 l1 l1 l1 • Which of the following has a transversal?

  11. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 DEFINITION • s 3, 4, 5 & 6 are INTERIOR ANGLES • s 1, 2, 7 & 8 are EXTERIOR ANGLES • ALTERNATE interior angles (AIA) - 2 non-adjacent interior angles on opposite sides of the transversal • Example: s 3 & 6 and s 4 & 5

  12. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 DEFINITION • s 3, 4, 5 & 6 are INTERIOR ANGLES • s 1, 2, 7 & 8 are EXTERIOR ANGLES • ALTERNATE exterior angles (AEA) - 2 non-adjacent exterior angles on opposite sides of the transversal • Example: s 1 & 8 and s 2 & 7

  13. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 DEFINITION • CONSECUTIVE interior angles (CIA) - 2 interior angles on the same side of the transversal • Example: s 3 & 5 and s 4 & 6

  14. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 DEFINITION • CORRESPONDING angles (CA) - 2 non-adjacent angles on the same side of the transversal such that one is an exterior angle and the other is an interior angle • Example: s 1 & 5, s 3 & 7, s 2 & 6, s 4 & 8

  15. Exercise *Identify the transversal . Identify each angle pair as AIA, AEA, CA, CIA or none of these. a. 13 & 5 b. 12 & 7 c. 10 & 7 d. 3 & 1 e. 3 & 16 f. 13 & 4 g. 10 & 1 h. 10 & 11 CA 1 2 3 4 AIA 5 6 7 8 none CA 9 11 12 10 AEA 15 14 13 16 none none CIA

  16. Parallel Line Properties

  17. 1 1 2 2 3 3 4 4 PROPERTIES of // Lines • What if our transversal is intersecting 2 // lines? • What relationships can we observe between: • CA? • AIA? • AEA? • CIA? 1 2 3 4 congruent 5 6 congruent 7 8 congruent supplementary

  18. Parallel Line Theorems • ÌF TWO LINES ARE PARALLEL… • CA Theorem - …then CORRESPONDING ANGLES are CONGRUENT. • AIA Theorem • - ….then ALTERNATE INTERIOR ANGLES are CONGRUENT. • AEA Theorem • - …then ALTERNATE EXTERIOR ANGLES are CONGRUENT. • CIA Theorem • - …then CONSECUTIVE INTERIOR ANGLES are SUPPLEMENTARY. * Prove algebraically.

  19. Practice (Source: DG by Serra) 2. a=b=c=54 b=d=65; a=c=115

  20. Practice (Source: DG by Serra) 4. 3. a=72; b=126

  21. Practice 5. 5x + 2 = 182 – 4x 9x = 180 x = 20 182 – 4x  102 102 = 4y + 2 y=25

  22. Practice 6. 7.

  23. Practice 8.

  24. Practice 9. What’s wrong with this picture? Explain.

  25. Practice 11. 10. m=38 m=125

  26. HOMEWORK a=102; b=78; c=f=58; d=122; e=26

  27. Parallel Line Properties(Part II)

  28. PROPERTIES of // Lines • Is the converse of the // Line Thm true? If 2 lines are cut by a transversal to form pairs of congruent CA, congruent AIA, and congruent AEA, then the lines are parallel.

  29. 4x – 12 3x + 2 b // by the Converse of AIA Thm Not // (CIAs are not supplementary) Not // What is b so that the 2 lines are parallel? 4x – 12 = 3x + 2 x = 14 4x – 12  44 b=136

  30. Practice

  31. Practice Determine which lines are parallel.

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