Geometry Presentation #2 Line and Angle Relationships & Classifying Polygons April 23, 2013 Math Block 4 Learning Objectives: Identify parallel, perpendicular, and skew lines, and angles formed by a transversal Identify and name polygons

Vocabulary perpendicular lines parallel lines skew lines adjacent angles vertical angles transversal

When lines, segments, or rays intersect, they form angles When lines, segments, or rays intersect, they form angles. If the angles formed by two intersecting lines measure 90°, the lines are perpendicular lines. Some lines in the same plane do not intersect at all. These lines are parallel lines. Segments and rays that are part of parallel lines are also parallel. Skew lines do not intersect, and yet they are also not parallel. They lie in different planes.

The symbol means “is parallel to The symbol means “is parallel to.” The symbol means “is perpendicular to.” Reading Math

Additional Example 1: Identifying Parallel, Perpendicular, and Skew Lines Tell whether the lines appear parallel, perpendicular, or skew. UV and YV The lines appear to intersect to form right angles. UV  YV XU and WZ The lines are in different planes and do not intersect. XU and WZ are skew. XY and WZ The lines are in the same plane and do not intersect. XY || WZ

Check It Out: Example 1A Tell whether the lines appear parallel, perpendicular, or skew. WX and XU WX  XU The lines appear to intersect to form right angles.

Check It Out: Example 1B Tell whether the lines appear parallel, perpendicular, or skew. WX and UV The lines are in different planes and do not intersect. WX and UV are skew.

Check It Out: Example 1C Tell whether the lines appear parallel, perpendicular, or skew. WX and ZY The lines are in the same plane and do not intersect. WX || ZY

Adjacent angles have a common vertex and a common side, but no common interior points. Angles 2 and 3 in the diagram are adjacent. Adjacent angles formed by two intersecting lines are supplementary Vertical angles are the opposite angles formed by two intersecting lines. Angles 1 and 3 in the diagram are vertical angles. Vertical angles have the same measure, so they are congruent.

Reading Math Angles with the same number of tick marks are congruent. The tick marks are placed in the arcs drawn inside the angles.

A transversal is a line that intersects two or more lines A transversal is a line that intersects two or more lines. Transversals to parallel lines form special angle pairs.

Additional Example 2A: Using Angle Relationships to Find Angle Measures Line n line p. Find the measure of the angle. 2 2 and the 130° angle are vertical angles. Since vertical angles are congruent, m2 = 130°.

Line n line p. Find the measure of the angle. Additional Example 2B: Using Angle Relationships to Find Angle Measures Line n line p. Find the measure of the angle. 3 Adjacent angles formed by two intersecting lines are supplementary. m3 + 130° = 180° –130° –130° Subtract 130° to isolate m3. m3 = 50°

Additional Example 2C: Using Angle Relationships to Find Angle Measures Line n line p. Find the measure of the angle. 4 Alternate interior angles are congruent. m4 = 130°.

Lesson Homework – Bring Answers to next class Tell whether the lines appear parallel, perpendicular, or skew. 1. AB and CD 2. EF and FH 3. AB and CG 4. In the figure on the right, line x || line y. Identify the measures of 2, 6, and 7. A. 70°, 110°, 70° B. 110°, 70°, 70° C. 70°, 70°, 110° 16

segment at a common point. This common point is a vertex of a polygon. Triangles and rectangles are examples of polygons. A polygon is a closed plane figure formed by three or more line segments. Each line segment forms a side of the polygon, and meets, but does not cross, another line segment at a common point. This common point is a vertex of a polygon. Side Vertex The polygon at left has six sides and six vertices. Vertices is plural for vertex. Remember!

Additional Example 1: Identifying Polygons Determine whether each figure is a polygon. If it is not, explain why not. A. B. The figure is a polygon. It is a closed figure with 4 line segments. The figure is not a polygon. It is not a closed figure.

Additional Example 1: Identifying Polygons Determine whether each figure is a polygon. If it is not, explain why not. C. D. The figure is not a polygon. The figure is not formed by line segments. The figure is not a polygon. There are line segments in the figure that intersect.

Polygons are classified by the number of sides and angles they have. Triangle 3 sides 3 angles Quadrilateral 4 sides 4 angles Pentagon 5 sides 5 angles Hexagon 6 sides 6 angles Heptagon 7 sides 7 angles Octagon 8 sides 8 angles Nonagon 9 sides 9 angles Decagon 10 sides 10 angles

Additional Example 2: Classifying Polygons Name each polygon. A. B. Octagon Quadrilateral

Check It Out: Example 2 Name each polygon. B. A. Quadrilateral Pentagon

A regular polygon is a polygon in which all sides are congruent and all angles are congruent.

Additional Example 3: Identifying and Classifying Regular Polygons Name each polygon and tell whether it is a regular polygon. If it is not, explain why not. B. A. The figure is a quadrilateral. It is an irregular polygon because all of the sides are not congruent. The figure is a regular quadrilateral. A regular quadrilateral is also called a square.

Lesson Homework – Bring Answers to next class Determine whether each figure is a polygon. If it in not, explain why not. Name each polygon. 1. 2. 3. 4. 5. Tell whether each figure above is a regular polygon. If it is not, explain why not. 25