Accelerated Algebra/Geometry Mrs. Crespo 2012-2013 Lesson 3-1 to 3-7 Accelerated Algebra/Geometry Mrs. Crespo 2012-2013

Lesson 3-1 Vocabulary & Key Concepts Postulate 3-1 Corresponding Angles Postulate If a transversal intersects two parallel lines, then corresponding angles are congruent. 1 l m 2

Lesson 3-1 Vocabulary & Key Concepts Theorem 3-1 Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. t 4 a b 3 2 1 5 6

Lesson 3-1 Vocabulary & Key Concepts Theorem 3-1 Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. t 4 a b 3 2 Theorem 3-2 Same Side Interior Angles Theorem If a transversal intersects two parallel lines, then same side interior angles are supplementary. 1 5 6

Lesson 3-1 Vocabulary & Key Concepts Theorem 3-3 Alternate Exterior Angles Theorem If a transversal intersects two parallel lines, then alternate exterior angles are congruent. t 4 a b 3 2 1 5 6

Lesson 3-1 Vocabulary & Key Concepts Theorem 3-3 Alternate Exterior Angles Theorem If a transversal intersects two parallel lines, then alternate exterior angles are congruent. t 4 a b 3 2 Theorem 3-4 Same Side Exterior Angles Theorem If a transversal intersects two parallel lines, then same side exterior angles are supplementary. 1 5 6

Lesson 3-1 Vocabulary & Key Concepts A transversal is a line that intersects two coplanar lines at two distinct points. l m

Lesson 3-1 Vocabulary & Key Concepts Alternate interior angles are non-adjacent interior angles that lie on opposite sides of the transversal. 5 6 l m 1 3 4 2 7 8

Lesson 3-1 Vocabulary & Key Concepts Same-side interior angles are interior angles that lie on the same side of the transversal. 5 6 l m 1 3 4 2 7 8

Lesson 3-1 Vocabulary & Key Concepts Corresponding angles are angles that lie on the same side of the transversal and in corresponding positions relative to the coplanar lines. 5 6 l m 1 3 4 2 7 8

Lesson 3-1 Vocabulary & Key Concepts A transversal is a line that intersects two coplanar lines at two distinct points. Alternate interior angles are non-adjacent interior angles that lie on opposite sides of the transversal. Same-side interior angles are interior angles that lie on the same side of the transversal. Corresponding angles are angles that lie on the same side of the transversal and in corresponding positions relative to the coplanar lines. 5 6 l m 1 3 4 2 7 8 Alternate Interior Angles <1 & <2 Same Side Interior Angles <1 & <4 Corresponding Angles <1 & <7

Alternate Interior Angles Lesson 3-1 Example 1 Applying Properties of Parallel Lines In the diagram of LRA, the black segments are runways. Compare <2 and the angle vertical to <1. Classify the angles. 3 1 2 X Alternate Interior Angles

by Corresponding < Post Lesson 3-1 Example 2 Finding Measures of Angles In the diagram l//m and p//q. Find m<1 and m<2. p q 8 6 42⁰ l 7 <1 and the 42⁰ angle are 4 2 1 = 42⁰ m corresponding angles 5 3 by Corresponding < Post m<1 + m<2 = 180⁰ by < Add’n Post

Lesson 3-1 Example 3 l a = 65⁰ c = 40⁰ m a + b + c = 180⁰ 75⁰ a = 65⁰ Alternate Interior <s Thm c = 40⁰ Alternate Interior <s Thm a + b + c = 180⁰ < Add’n Post 65 + b + 40 = 180⁰ Substitution b = 75⁰ Subtraction

Lesson 3-2 Vocabulary and Key Concepts Postulate 3-2 Converse of the Corresponding Angles Postulate If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel. 1 2 l m l // m

Lesson 3-2 Vocabulary & Key Concepts Theorem 3-5 Converse of the Alternate Interior Angles Theorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. t then l // m l m 1 2

Lesson 3-2 Vocabulary & Key Concepts Theorem 3-5 Converse of the Alternate Interior Angles Theorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. t then l // m l m 1 4 2 Theorem 3-6 Converse of the Same Side Interior Angles Theorem If two lines and a transversal form same side interior angles that are supplementary, then the two lines are parallel. then l // m

Lesson 3-2 Vocabulary & Key Concepts Theorem 3-7 Converse of the Alternate Exterior Angles Theorem If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. t then l // m 5 l m 3

Lesson 3-2 Vocabulary & Key Concepts Theorem 3-7 Converse of the Alternate Exterior Angles Theorem If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. t then l // m 5 6 l m Theorem 3-8 Converse of the Same Side Exterior Angles Theorem If two lines and a transversal form same side exterior angles that are supplementary, then the two lines are parallel. 3 then l // m

<3 and <2 are supplementary Lesson 3-2 Example 1 Using Postulate 3-2 Which lines if any, must be parallel if <3 and <2 are supplementary? Justify. 3 E C <3 and <2 are supplementary D 1 4 K <4 and <2 are supplementary 2 By Congruent Supplements Thm By Converse of Corresponding <s Post. ray EC // ray DK

Lesson 3-2 Example 2 l m Using Algebra Find the value of x for which l//m. The labeled angles are l alternate interior angles. (14+3x)⁰ If l//m, the alternate interior <s are (5x-66)⁰ m congruent. Thus, equal. 5x – 66 = 14 + 3x 5x = 80 + 3x 2x = 80 x = 40

Lesson 3-3 Key Concepts a a//b b c t m m//n n n l m a Theorem 3-9 If two lines are parallel to the same line, then they are parallel to each other. b a a//b c t Theorem 3-10 In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. m n m//n a n 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. m l

Lesson 3-3 Example 2 k l m Using Theorem 3-11 Write a paragraph proof. Given: In a plane, k//l and k//m. Also, m<1 = 90⁰. k m l Prove: The transversal is perpendicular to line m. Since m<1 = 90, the transversal is perp. to line k. Since k//m, by Theorem 3-11 the transversal is also perpendicular to line m.

Lesson 3-4 Vocabulary & Key Concepts Theorem 3-12 Triangle Angle Sum Theorem The sum of the measures of the angles of a triangle is 180⁰. m<A + m<B + m<C = 180⁰. Interior < Theorem 3-13 Triangle Exterior Angle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of its remote interior angles. Exterior < 2 1 3 Interior <

Lesson 3-4 Vocabulary & Key Concepts ANGLES An acute triangle has three acute angles. A right triangle has one right angle. An obtuse triangle has one obtuse angle. An equiangular triangle has three congruent angles. SIDES An equilateral triangle has three congruent sides. An isosceles triangle has at least two congruent sides. A scalene triangle has no congruent sides.

Lesson 3-4 Vocabulary & Key Concepts An exterior angle of a polygon is an angle formed by a side and an extension of an adjacent side. Remote interior angles are two non-adjacent interior angles corresponding to each exterior angle of a triangle. Exterior Angle 1 3 2 Remote Interior Angles

Lesson 3-4 Example 1 Applying the Triangle Angle-Sum Theorem In triangle ABC, <ACB is a right angle; segment CD perpendicular to segment AB. Find the values of a and c. Find c m<ACB =90⁰ Definition of a Right < c + 70 = 90 Angle Addition Postulate c = 20 Subtract 70 from each side. Find a a + m<ADC + c = 180 Triangle Angle Sum Theorem m<ADC = 90 Def’n of Perpendicular Lines c + 90 + 20 = 180 Sub. 90 for m<ADC & 20 for c. a + 110 = 180 Simplify. a = 70 Subtract 110 from each side.

Lesson 3-4 Example 2 Back x⁰ (180⁰ - x)⁰ Arm Applying the Triangle Exterior Angle Theorem Explain what happens to the angle formed by the back of the chair and the armrest as you make a lounge chair recline more. Back x⁰ x⁰ (180⁰ - x)⁰ Arm (180⁰ - x)⁰ The exterior angle and the angle formed by the back of the chair and the armrest are adjacent angles, which together form a straight angle. As one measure increases, the other decreases. The angle formed by the back of the chair and the armrest increases as you make a lounge chair recline more.

Lesson 3-5 Vocabulary & Key Concepts Theorem 3-14 Polygon Angle-Sum Theorem The sum of the measures of the angles of an n-gon = (n-2)180⁰. Theorem 3-15 Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex is 360⁰. 4 5 1 3 2 Pentagon m<1 + m<2 + m<3 + m<4 + m<5 = 360⁰.

Lesson 3-5 Vocabulary & Key Concepts A polygon is a closed plane figure with at least three sides that are segments. The sides intersect only at their endpoints, and no two adjacent sides are collinear. A C D E B E B C A D A B D C E Not a polygon; two sides intersect between endpoints & two adjacent sides are collinear. A polygon. Not a polygon; not a closed figure

Lesson 3-5 Vocabulary & Key Concepts A polygon is a closed plane figure with at least three sides that are segments. The sides intersect only at their endpoints, and no two adjacent sides are collinear. YES!!!

Lesson 3-5 Vocabulary & Key Concepts Polygons are either convex or concave. A convex polygon does not have diagonal points outside of the polygon. A concave polygon has at least one diagonal with points outside of the polygon. An equilateral polygon has all sides congruent. An equiangular polygon has all angles congruent. A regular polygon is both equilateral and equiangular. R D Y T A P M S W Q K G

Lesson 3-5 Vocabulary & Key Concepts Convex line segments connecting any two points on the shape lie entirely inside the shape Concave at least, one line segment connecting any two points on the shape pass outside the shape

Lesson 3-5 Example 1 Classify the polygon by its sides. Identify it as convex or concave. Number of sides: 12 Name: Dodecagon Convex? No. Concave? Yes.

Lesson 3-5 Example 2 Find the sum of the measures of the angles of a decagon. Number of sides: 10 So, n = 10 Sum = (n-2)180 Polygon Angle Sum Theorem = (10-2)180 Substitute 10 for n = 8(180) Subtract = 1440 Simplify

m<X + m<Y + m<Z + m<W = (4-2)180=360. Lesson 3-5 Example 3 Use the Polygon Angle Sum Theorem. Find m<x in quadrilateral XYZW. Number of sides: 4 So, n = 4 Y Z X W 100⁰ m<X + m<Y + m<Z + m<W = (4-2)180=360. Polygon Angle Sum Theorem m<X + m<Y + 90 + 100 = 360. m<X + m<Y = 170. m<X + m<X = 170. 2m<X = 170. m<X = 85.

Lesson 3-5 Example 4 Applying Theorem 3-15. A regular hexagon is inscribed in a rectangle . Explain how you know that all angles labeled <1 have equal measures. The hexagon is regular, so all its angles are congruent. An exterior angle is the supplement of a polygon’s angle because they are adjacent angles that form a straight angle. Because supplements of congruent angles are congruent, all the angles marked <1 have equal measures. 2 1

where m is the slope and b is the y-intercept. Lesson 3-6 Vocabulary Ex. y=2x+3 The slope-intercept form of a linear equation is 𝒚=𝒎𝒙+𝒃 where m is the slope and b is the y-intercept. Ex. 2x-y=1 The standard form of a linear equation is 𝑨𝒙+𝑩𝒚=𝑪 Ex. y+2=2(x-1) The point-slope form for a non-vertical line is 𝒚− 𝒚 𝟏 =𝒎(𝒙− 𝒙 𝟏 ) where m is the slope and 𝒙 𝟏 and 𝒚 𝟏 are point coordinates.

Lesson 3-6 Quickie y x Intercepts are points of intersection where the graph crosses either or both the axes. y-intercept x-intercept

Lesson 3-6 Example 1 Graphing Lines Using Intercepts. Use the x-intercept and y-intercept to graph 5x-6y = 30. Find the x-intercept. Sub 0 for y. Solve for x. 5x – 6y = 30 5x – 6 (0) = 30 5x – 0 = 30 5x = 30 x = 6 , the x-intercept. As a point, it is (6,0). Find the y-intercept. Sub 0 for x. Solve for y. 5x – 6y = 30 5(0) – 6y = 30 0 – 6y = 30 -6y = 30 y = -5 , the y-intercept. As a point, it is (0,-5).

Lesson 3-6 Example 1 Graphing Lines Using Intercepts. y x Graphing Lines Using Intercepts. Use the x-intercept and y-intercept to graph 5x-6y = 30. The x-intercept. As a point, it is (6,0). The y-intercept. As a point, it is (0,-5).

Lesson 3-6 Example 2 Transforming to Slope-Intercept Form. Transform -6x +3y =12 to slope intercept form. Then, graph. -6x + 3y = 12 3y = 6x + 12 (3y)/3 = (6x)/3 + (12)/3 y = 2x + 4 The y-intercept is 4 and the slope is 2. Plot. Graph the y-intercept; as a point it is (0,4). With slope, 2 is 2/1. So, up 2, right 1. Connect the two points.

Lesson 3-6 Example 2 Transforming to Slope-Intercept Form. Transform -6x +3y =12 to slope intercept form. Then, graph. y x Plot. Graph the y-intercept; as a point it is (0,4). With slope, 2 is 2/1. So, up 2, right 1. Connect the two points.

Lesson 3-6 Examples 3 & 4 on the board. Lesson 3-7 on the board.

PowerPoint by Mrs. Crespo for Accelerated Algebra/Geometry 2012-2013 Acknowledgement Prentice Hall Mathematics Geometry by Bass, Charles, Hall, Johnson and Kennedy 2007 PowerPoint by Mrs. Crespo for Accelerated Algebra/Geometry 2012-2013