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Geometry CHAPTER 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Basic Geometric Figures Perimeter Area Circles Volume and Surface Area Relationships Between Angle Measures Congruent Triangles and Properties of Parallelograms 8.8 Similar Triangles Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 2 8.6 Relationships Between Angle Measures OBJECTIVES a Identify complementary and supplementary angles and find the measure of a complement or a supplement of a given angle. b Determine whether segments are congruent and whether angles are congruent. c Use the Vertical Angle Property to find measures of angles. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 3 8.6 Relationships Between Angle Measures OBJECTIVES d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 4 8.6 Relationships Between Angle Measures COMPLEMENTARY ANGLES Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 5 8.6 Relationships Between Angle Measures Identify complementary and supplementary angles a and find the measure of a complement or a supplement of a given angle. If two angles are complementary, each is an acute angle. When complementary angles are adjacent to each other, they form a right angle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 6 8.6 a Relationships Between Angle Measures Identify complementary and supplementary angles and find the measure of a complement or a supplement of a given angle. EXAMPLE 2 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 7 8.6 Relationships Between Angle Measures SUPPLEMENTARY ANGLES Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 8 8.6 a Relationships Between Angle Measures Identify complementary and supplementary angles and find the measure of a complement or a supplement of a given angle. EXAMPLE 4 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 9 8.6 Relationships Between Angle Measures Determine whether segments are congruent and b whether angles are congruent. Congruent figures have the same size and shape. They fit together exactly. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 10 8.6 Relationships Between Angle Measures CONGRUENT SEGMENTS Two segments are congruent if and only if they have the same length. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 11 8.6 Relationships Between Angle Measures Determine whether segments are congruent and b whether angles are congruent. EXAMPLE 6 Which pairs of segments are congruent? Use a ruler. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 12 8.6 Relationships Between Angle Measures CONGRUENT ANGLES Two angles are congruent if and only if they have the same measure. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 13 8.6 Relationships Between Angle Measures Determine whether segments are congruent and b whether angles are congruent. EXAMPLE 8 Which pairs of angles are congruent? Use a protractor. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 14 8.6 Relationships Between Angle Measures Determine whether segments are congruent and b whether angles are congruent. If two angles are congruent, then their supplements are congruent and their complements are congruent. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 15 8.6 Relationships Between Angle Measures VERTICAL ANGLES Two nonstraight angles are vertical angles if and only if their sides form two pairs of opposite rays. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 16 8.6 Relationships Between Angle Measures THE VERTICAL ANGLE PROPERTY Vertical angles are congruent. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 17 8.6 Relationships Between Angle Measures Use the Vertical Angle Property to find measures c of angles. EXAMPLE 9 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 18 8.6 Relationships Between Angle Measures Use the Vertical Angle Property to find measures c of angles. EXAMPLE 9 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 19 8.6 Relationships Between Angle Measures TRANSVERSAL A transversal is a line that intersects two or more coplanar lines in different points. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 20 Relationships Between Angle Identify pairs of corresponding angles, interior angles, Measures 8.6 and alternate interior angles and apply properties of d transversals and parallel lines to find measures of angles. Corresponding Angles Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 21 Relationships Between Angle Identify pairs of corresponding angles, interior angles, Measures 8.6 and alternate interior angles and apply properties of d transversals and parallel lines to find measures of angles. Interior Angles Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 22 Relationships Between Angle Identify pairs of corresponding angles, interior angles, Measures 8.6 and alternate interior angles and apply properties of d transversals and parallel lines to find measures of angles. Alternate Interior Angles Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 23 8.6 d Relationships Between Angle Measures Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. EXAMPLE 10 Identify all pairs of corresponding angles, all interior angles, and all pairs of alternate interior angles. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 24 8.6 d Relationships Between Angle Measures Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. EXAMPLE 10 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 25 Relationships Between Angle Identify pairs of corresponding angles, interior angles, Measures 8.6 and alternate interior angles and apply properties of d transversals and parallel lines to find measures of angles. Properties of Parallel Lines 1. If a transversal intersects two parallel lines, then the corresponding angles are congruent. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 26 Relationships Between Angle Identify pairs of corresponding angles, interior angles, Measures 8.6 and alternate interior angles and apply properties of d transversals and parallel lines to find measures of angles. 2. If a transversal intersects two parallel lines, then the alternate interior angles are congruent. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 27 Relationships Between Angle Identify pairs of corresponding angles, interior angles, Measures 8.6 and alternate interior angles and apply properties of d transversals and parallel lines to find measures of angles. 3. In a plane, if two lines are parallel to a third line, then the two lines are parallel to each other. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 28 Relationships Between Angle Identify pairs of corresponding angles, interior angles, Measures 8.6 and alternate interior angles and apply properties of d transversals and parallel lines to find measures of angles. 4. If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 29 Relationships Between Angle Identify pairs of corresponding angles, interior angles, Measures 8.6 and alternate interior angles and apply properties of d transversals and parallel lines to find measures of angles. 5. If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 30 8.6 d Relationships Between Angle Measures Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. EXAMPLE 11 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 31 8.6 d Relationships Between Angle Measures Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. EXAMPLE 11 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 32 8.6 d Relationships Between Angle Measures Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. EXAMPLE 12 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 33