Polygons Essential Question: Why is it important to understand the properties of two-dimensional figures, such as triangles and quadrilaterals?

Angle Relationships Target: Classify and identify angles and find missing measures.

An angle has two sides that share a common endpoint called a vertex. Angle Definitions An angle has two sides that share a common endpoint called a vertex. Angles are measures in units called degrees. How many degrees are in a circle? Congruent angles have the same measure. Naming Angles Use the vertex as the middle letter and a point from each side. The symbol for angle is . LMN or NML Use only the vertex. M Use a number. 1 L M 1 N

Types of Angles

Angle Practice Name each angle in two different ways. Classify each angle as acute, obtuse, right, or straight. ABC CBA B 1 straight MNO ONM N 2 right PQR RQP Q 3 acute STU UTS T 4 obtuse Angle Practice

Pairs of Angles The symbol  is used to represent “congruent.” Adjacent Angles are two angles that share a vertex and a common side and do not overlap. Vertical Angles are angles formed when two lines intersect – two pairs of congruent opposite angles are created. v Adjacent Angles v Vertical Angles Complementary Angles are two angles whose measures add up to 90. Supplementary Angles are two angles whose measures add up to 180. Complementary Angles Supplementary Angles The symbol  is used to represent “congruent.” 1  2 is read as angle 1 is congruent to angle 2. Pairs of Angles

Problem Solving With Pairs of Angles If ∠A and ∠B are complementary and the measure of ∠A is 86°, what is the measure of ∠B? 4° What is the measure of ∠C if ∠C and ∠D are supplementary and the measure of ∠D is 97°? 83° Determine whether the statement is true or false. If the statement is true, draw a diagram to support it. If the statement is false, explain why. An obtuse angle and an acute angle are always supplementary. FALSE. Complementary angles must be acute. TRUE Problem Solving With Pairs of Angles

Lines in a plane that never intersect are parallel lines Lines in a plane that never intersect are parallel lines. When two parallel lines are intersected by a third line, this line is called a transversal. If a pair of parallel lines is intersected by a transversal, these pairs of angles are congruent. Alternate interior angles are on opposite sides of the transversal and inside the parallel lines. 3  5 , 4  6 Alternate exterior angles are on opposite sides of the transversal and outside the parallel lines. 1  7 , 2  8 Corresponding angles are in the same position on the parallel lines in relation to the transversal. 1  5 , 2  6 3  7 , 4  8 Angle Relationships

Using Angle Relationships Classify each pair of angles shown. 1 and 5 corresponding 3 and 5 alternate interior 6 and 4 7 and 1 alternate exterior In the figure, if m2 = 74°, find each measure. m8 74° m6 m4 m1 106° Using Angle Relationships

Triangles Target: Classify triangles and find missing angle measures.

A triangle is a figure with three sides and three angles A triangle is a figure with three sides and three angles. The symbol for triangle is △. The sum of the measures of the angles of a triangle is 180°. In △ABC, if mA = 25° and mB = 108°, what mC? Add up the measures given and subtract from 180. 47° Find the missing measures in the given triangles. Angles of Triangles

Every triangle has at least two acute angles Every triangle has at least two acute angles. One way to classify angles is to use the third angle. Another way to classify angles is by their sides. Sides with the same length are congruent segments. The tick marks on the sides of the triangles indicate that those sides are congruent. Classify Triangles

Practice with Triangles Find the missing angle measure. Classify each triangle by its angles and its sides. 44 134 45 Acute, equilateral Right, scalene Acute, isosceles Practice with Triangles

Triangle ABC is formed by two parallel lines and two transversals Triangle ABC is formed by two parallel lines and two transversals. Find the measure of each interior angle A, B, and C of the triangle. With your group, discuss this problem and how you might go about solving it. You may want to look back in your notes about parallel lines and transversals. mA = 61° mB = 72° mC = 47° Challenge!

Quadrilaterals Target: Classify quadrilaterals and find missing angle measures.

Angles of a Quadrilateral A quadrilateral has four sides and four angles. The sum of the measures of the angles of a quadrilateral is 360°. Find the missing angle in each quadrilateral. 58° 161° Angles of a Quadrilateral

Classifying Quadrilaterals The red arcs show congruent angles. The red square corner indicates a perpendicular line, forming a right angle. Classifying Quadrilaterals

Practice with Quadrilaterals Find the missing angle. Classify each quadrilateral. 100 135 65 square rectangle parallelogram Practice with Quadrilaterals quadrilateral trapezoid rhombus

Polygons and Angles Target: Find the sum of the angle measures of a polygon and the measure of an interior angle of a regular polygon.

A polygon is a simple, closed figure formed by three or more straight line segments. A simple figure does not have lines that cross each other. You have drawn a closed figure when your pencil ends up where it started. Polygons

Polygon Classification Polygons are classified by the number of sides it has. An equilateral polygon has all sides congruent. A polygon is equiangular if all of its angles are congruent. A regular polygon is equilateral and equiangular, with all sides and angles congruent. Polygon Classification

Finding Interior Angles The sum of the measures of the angles of a triangle is 180°. You can use this relationship to find the measures of the angles of polygons. With your partner, use diagonals to find the sum of the interior angles of several different polygons. Use the worksheet provided. Interior Angle Sum of a Polygon The sum of the measures of the angles of a polygon is (n – 2)180, where n represents the number of sides. S = (n – 2)180 Finding Interior Angles

Interior Angle Practice Using what you know about the sum of interior angles, find the value of each variable. x = 83, y = 74 x = 128 x = 20 Interior Angle Practice