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Math 310

Math 310. Section 9.3 More on Angles. Linear Pair. Def Two angles forming a line are called a linear pair. Ex. Not a linear pair: <ABC & <FDE. Linear pairs: <ABC & <DBC <BDE & <FDE. Question. What can we say about the sum of the measures of the angles of a linear pair?.

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Math 310

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  1. Math 310 Section 9.3 More on Angles

  2. Linear Pair Def Two angles forming a line are called a linear pair.

  3. Ex. Not a linear pair: <ABC & <FDE Linear pairs: <ABC & <DBC <BDE & <FDE

  4. Question What can we say about the sum of the measures of the angles of a linear pair?

  5. Vertical Angles Def When two lines intersect, four angles are created. Taking one of the angles, along with the other angle which is not its linear pair, gives you vertical angles. (ie it is the angle “opposite” of it)

  6. Ex. Vertical angles: <ABC & <EBD <CBE & <DBA

  7. Vertical Angle Theorem Thrm Vertical angles are congruent.

  8. Ex. If m<ABC = 95° find the other three angle measures. m<EBD = 95° m<CBE = 85° m<DBA = 85°

  9. Supplementary Angles Def Supplementary angles are any two angles whose sum of their measures is 180°.

  10. Ex. <ABC & <CBE <ABC & <FED <ABC & <BEG <DEB & <FED <DEB & <CBE <DEB & <BEG <GEF & <FED <GEF & <CBE <GEF & <BEG Given: <ABC is congruent to <FEG Find all pairs of supplementary angles.

  11. Complementary Angles Def Complementary angles are any two angles whose sum of their measures is 90°.

  12. Ex. Given: ray BC is perpendicular to line AE. Name all pairs of complementary angles. <CND & <DBE

  13. 25° 65° 65° Ex. Name all pairs of complementary angles. <ABC & <GHI <DEF & <GHI

  14. Transversal Def A line, crossing two other distinct lines is called a transversal of those lines.

  15. Ex. Name two lines and their transversal. Lines: JK & QO Transversal: OK

  16. Transversals and Angles Given two lines and their transversal, two different types of angles are formed along with 3 different pairs of angles: • Interior angles • Exterior angles • Alternate interior angles • Alternate exterior angles • Corresponding angles

  17. Interior Angles <JKO <MKO <QOK <NOK

  18. Exterior Angles <JKL <MKL <QOP <NOP

  19. Alternate Interior Angles <JKO & <NOK <MKO & <QOK

  20. Alternate Exterior Angles <JKL & <NOP <MKL & <QOP

  21. Corresponding Angles <JKL & <QOK <MKL & <NOK <QOP & <JKO <NOP & <MKO

  22. Parallel Lines and Transversals Thrm If any two distinct coplanar lines are cut by a transversal, then a pair of corresponding angles, alternate interior angles, or alternate exterior angles are congruent iff the lines are parallel.

  23. Ex. Given: Lines AB and GF are parallel. Name all congruent angles. <ABC & <GFB <DBC & <EFB <GFH & <ABF <EFH & <DBF <ABC & <EFH <DBC & <GFH <DBF & <GFB <ABF & <EFB <ABC & <GFB

  24. Triangle Sum Thrm The sum of the measures of the interior angles of a triangle is 180°.

  25. Angle Properties of a Polygon Thrm • The sum of the measures of the interior angles of any convex polygon with n sides is 180n – 360 or (n – 2)180. • The measure of a single interior angle of a regular n-gon is (180n – 360)/n or (n – 2)180/n.

  26. Ex. What is the sum of the interior angles of a heptagon? A dodecagon? Heptagon: (7 – 2)180° = (5)180° = 900° Dodecagon: (10 – 2)180° = (8)180° = 1440°

  27. Exterior Angle Theorem Thrm The sum of the measures of the exterior angles (one at each vertex) of a convex polygon is 360°.

  28. Proof Given a convex polygon with n sides and vertices, lets say the measure of each interior angles is x1, x2, …., xn. Then the measure of one exterior angle at each vertices is 180 – xi. Adding up all the exterior angles: (180 – x1) + (180 – x2) + … + (180 – xn) = 180n – (x1 + x2 +…+ xn) = 180n – (180n – 360 ) = 180n – 180n + 360 = 360

  29. Ex. Pg 610 – 12a Pg 610 - 7

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