LESSONS 1-5 TO 1-7 Accelerated Algebra/Geometry Mrs. Crespo

Lesson 1-5: Measuring Segments Recap  Postulate 1-5: Ruler Postulate  Postulate 1-6: Segment Addition Postulate (AB+BC=AC)  Definition of Coordinate, Congruent Segments and Midpoint. AB C AB C

Lesson 1-5: Examples  Example 1 Comparing Segment Lengths  Example 2 Using Addition Segment Postulate If AB=25, find x. Then, find AN and NB. AN B 2x-6 x+7 AN + NB = AB (2x-6) +( x+7) = 25 3x + 1 = 25 3x = 24 x = 24/3 x = 8 AN = 2x – 6 = 2(8) – 6 = 16 – 6 = 10 NB = x + 7 = 8 +7 = 15

R M T 5x+9 8x-36 M Lesson 1-5: Examples RM = MT 5x + 9 = 8x – 36 5x – 8x = -36 – 9 -3x = -45 x = -45/-3 x = 15 RM = 5x + 9 = 5(15) + 9 = = 84 MT = 8x – 36 = 8(15) – 36 = 120 – 36 = 84 RT = RM + MT = = 168  Example 3 Using Midpoint M is the midpoint of segment RT. Find RM, MT, and RT.

Vocabulary and Key Concepts  Postulate 1-7: Protractor Postulate  Postulate 1-8: Angle Addition Postulate (m<AOB + m<BOC = m<AOC)  Definition of Angle Formed by two rays with the same endpoint. T Q B Lesson 1-6: Measuring Angles A B C O 1

Vocabulary and Key Concepts  Acute Angle: measures between 0 0 and 90 0  Right Angle: measures exactly 90 0  Obtuse Angle: measures between 90 0 and  Straight Angle: measures exactly  Congruent angles: two angles with the same measure x0x0 Lesson 1-6: Measuring Angles x0x0 ACUTE ANGLE RIGHT ANGLE 0 < x < 90 0 x = 90 0 x0x < x < OBTUSE ANGLE x0x0 x = STRAIGHT ANGLE

 Example 1 Naming Angles Lesson 1-6: Examples NN ame can be the number between the sides of the angle. <3 A G 3 C <CGA <G <AGC NN ame can be the vertex of the angle. NN ame can be a point on one side, the vertex, and a point on the other side of the angle. or

 Example 2 Measuring and Classifying Angles Lesson 1-6: Examples  Find the measure of each <AOC. m<AOC =  Classify as acute, obtuse, or straight. A C O B O C B A 60 0 m<AOC =150 0

 Example 3 Using the Angle Addition Postulate Lesson 1-6: Examples  Suppose that m<1=42 and m<ABC=88. Find m<2 m<1 + m<2 = m<ABC 42 + m<2 = 88 m<2 = m<2 = 46 0 B A C 1 2

 Example 4 Identifying Angle Pairs Lesson 1-6: Examples  In the diagram, identify pairs of numbered angles as: Complementary angles form 90 0 angles. <3 and < Supplementary angles form angles. Vertical angles form an “X”. <1 and <2<2 and <3 <1 and <3

 Example 5 Making Conclusions From A Diagram Lesson 1-6: Examples  Can you make each conclusion from a diagram? 3 m<BCA + m<DCA = <B and <ACD are supplementary. B A D C <A <C segment AB segment BC

Vocabulary  Construction is using a straightedge and a compass to draw a geometric figure.  A straightedge is a ruler with no markings on it.  A compass is a geometric tool used to draw circles and parts of circles called arcs. Lesson 1-7: Basic Construction

B A D C Vocabulary  Perpendicular lines are two lines that intersect to form right angles. Lesson 1-7: Measuring Angles  A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby, bisecting the segment into two congruent segments.  An angle bisector is a ray that divides an angle into two congruent coplanar angles. N L K J

T  Example 1 Constructing Congruent Segments Lesson 1-7: Examples  Construct segment TW congruent to segment KM. STEP 1: Draw a ray with endpoint T. KM STEP 2: Open the compass the length of segment KM. W STEP 3: With the same compass setting, put the compass point on point T. Draw an arc that intersects the ray. Label the point of intersection W.

 Example 2 Constructing Congruent Angles Lesson 1-7: Examples  Construct <Y so that <Y is congruent to <G. Y 75 0 G E F Z <Y <G 1.Draw a ray with endpoint Y. 2.With the compass point on G, draw an arc that intersects both sides of <G. Label the points of intersection E and F. 3.With the same compass setting, put the compass point on point Y. Draw an arc that intersects the ray. Label the point of intersection Z. 4.Open the compass to the length EF. Keeping the same compass setting, put the compass on point Z. Draw an arc that intersects with the arc previously. Label the point of intersection X. 5.Draw ray YX to complete <Y. X

 Example 3 Constructing The Perpendicular Bisector Lesson 1-7: Examples  Given segment AB. Construct line XY so that line XY is perpendicular to segment AB at the midpoint M of segment AB. 1.Put the compass point on point A and draw a long arc. Be sure the opening is greater than half of AB. 2.With the same compass setting, put the compass point on point B and draw another long arc. Label the points where the two arcs intersect as an X and Y. 3.Draw line XY. The point of intersection of segment AB and line XY is M, the midpoint of segment AB. A B X Y M

 Example 4 Finding Angle Measures Lesson 1-7: Examples  Line WR bisects <AWB so that m<AWR=x and m<BWR=4x-48. Find m<AWB. m<AWR = m<BWR x = 4x – 48 -3x = -48 x = 16 R W A B 4x – 48 x m<AWR = x = 16 m<BWR = 4x – 48 = 4(16) – 48 = 64 – 48 = 16 So, m<AWB = m<AWR + m<BWR = = 32

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Reference Textbook: Prentice Hall Mathematics GEOMETRY by Bass, Charles, Hall, Johnson, Kennedy PowerPoint Created by Mrs. Crespo Accelerated Algebra/Geometry Mrs. Crespo