Warm Up- JUSTIFY WHAT THEOREM OR POSTULATE YOU ARE USING! 2x - 75 Find the value of x for which a ll t Find the value of y. Find the measurements of the angles. 2x - 20 x +35 50 70 x y 2x y y – 50

Homework Answers 11) a//b if two lines and a trans. Form SSIA that are suppl, then the two lines are // Line BE and CG are parallel- converse of corresponding angle postulate 12) a//b if two lines and a trans. Form SSEA that are suppl, then the two lines are // Segment CA and segment HR- Converse of corr. Angle post None Segment JO is parallel to LM- if two lines and a transversal form SSIA that are supp, then the lines are parallel a//b converse of corr angle post a//b- Conv of AIAT l//m Conv of Corr Angle Post Segment PQ and segment ST- Converse of Alt. Int. Angle Thm a//b if two lines and a trans. Form AEA that are congruent, then two lines are // 30 50 a//b Conv of Corr Angle Post 59 31 l//m conv of AIA thm 10) a//b if two lines and a transversal form SSIA that are supp, then the lines are // a. <1 b. <1 c. <2 d. <3 e. Converse of Corr Angles

Theorem 3-9: If two lines are parallel to the same line, then they are parallel to each other. Draw a diagram for this theorem.

Prove the previous theorem. Given: j // k and r // k Prove: j // r 1 2 3

Perpendicular Line Theorems Draw a diagram that represents each theorem Theorem 3-10: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Theorem 3-11: In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

Triangle Angle-Sum Theorem The sum of the measures of the angles of a triangle is 180. m<1 + m<2 + m<3 = 180 Example: Find the value of x, y and z. 2 1 3 21 39 x y z 65

Draw a triangle that represents each classification. Equiangular All angles are congruent Right One right angle Acute All angles acute Obtuse One obtuse angle Equilateral All sides congruent Isosceles At least two sides congruent Scalene No sides congruent

Vocabulary Exterior Angle 1 3 2 Remote Interior Angles

Triangle Exterior Angle Theorem What is the relationship between the three angles- measure them and draw a conclusion. 2 1 3 m<1 = m<2 + m<3 The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

Polygon Is a closed plane figure with at least three sides that are segments. The sides intersect at their endpoints and no adjacent sides are collinear. When naming a polygon, start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction

Polygon Classification Polygons are classified as convex or concave Convex: has no diagonal with points outside of the polygon Concave: has at least one diagonal with points outside the polygon Polygons can be classified by the number of sides it has Ex. 3 sides- triangle, 4 sides quadrilateral, 5 sides pentagon

Pairs

The Sum of Polygon Angle Measures Sketch Polygons with each number of sides. Divide each polygon into triangles by drawing all diagonals that are possible from one vertex. Find the sum of the measures of each polygon using triangle angle sum theorem. Look for patterns in the table, write a rule for the sum of the measures of the angles of an n-gon. Polygon Number of Sides Number of Triangles Formed Sum of the Interior Angle Measures 4 5 6 7 8

Polygon Angle Sum Theorem The sum of the interior angle measure of a convex polygon with n sides is (n – 2)180°