Math 310 Section 9.3 More on Angles

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

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

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

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)

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

Vertical Angle Theorem Thrm Vertical angles are congruent.

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

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

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

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

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

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

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

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

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 Interior angles Exterior angles Exterior angles Alternate interior angles Alternate interior angles Alternate exterior angles Alternate exterior angles Corresponding angles Corresponding angles

Interior Angles <JKO <MKO <QOK <NOK

Exterior Angles <JKL <MKL <QOP <NOP

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

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

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

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.

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

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

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 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. The measure of a single interior angle of a regular n-gon is (180n – 360)/n or (n – 2)180/n.

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°

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

Proof Given a convex polygon with n sides and vertices, lets say the measure of each interior angles is x 1, x 2, …., x n. Then the measure of one exterior angle at each vertices is 180 – x i. Adding up all the exterior angles: (180 – x 1 ) + (180 – x 2 ) + … + (180 – x n ) = 180n – (x 1 + x 2 +…+ x n ) = 180n – (180n – 360 ) = 180n – 180n = 360

Ex. Pg 610 – 12a Pg