Continued Proofs Transversals and Special Angles
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Unit 3 Angles and Transversals Unit 3: This unit Introduces Transversals, and angles based on transversals, including corresponding angles, Same Side Interior Angles, Alternate Exterior and Alternate Interior Angles, and the unique properties of Perpendicular Transversals. 125 2 3 4 5 X+ 15 7 8
• • • Standards SPI's taught in Unit 3: SPI 3108. 1. 1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and space. SPI 3108. 1. 4 Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems. SPI 3108. 2. 1 Analyze, apply, or interpret the relationships between basic number concepts and geometry (e. g. rounding and pattern identification in measurement, the relationship of pi to other rational and irrational numbers) SPI 3108. 4. 2 Define, identify, describe, and/or model plane figures using appropriate mathematical symbols (including collinear and non-collinear points, lines, segments, rays, angles, triangles, quadrilaterals, and other polygons). CLE (Course Level Expectations) found in Unit 3: CLE 3108. 3. 1 Use analytic geometry tools to explore geometric problems involving parallel and perpendicular lines, circles, and special points of polygons. CLE 3108. 4. 1 Develop the structures of geometry, such as lines, angles, planes, and planar figures, and explore their properties and relationships. CFU (Checks for Understanding) applied to Unit 3: 3108. 4. 8 Apply properties and theorems about angles associated with parallel and perpendicular lines to solve problems. 3108. 4. 21 Use properties of and theorems about parallel lines, perpendicular lines, and angles to prove basic theorems in Euclidean geometry (e. g. , two lines parallel to a third line are parallel to each other, the perpendicular bisectors of line segments are the set of all points equidistant from the endpoints, and two lines are parallel when the alternate interior angles they make with a transversal are congruent).
Transversal • A transversal is a line which cuts across two or more coplanar lines –they may or may not be parallel • Transversals create special angle pairs which have properties that we can use to solve their measure of degree Transversal Parallel Lines are indicated by arrows
Corresponding Angles • Corresponding Angles: These are angles which occupy corresponding (or matching) positions on two different lines • Angle 1 corresponds with Angle 5 • Which angle corresponds with angle 4? • Which angle corresponds with angle 2? 1 2 3 4 5 6 7 8
Corresponding Angles on Parallel Lines • Corresponding Angles Postulate – If a transversal intersects 2 parallel lines, then corresponding angles are congruent • If the measure of angle 3 is 75 degrees, what is the measure of angle 2? (Vertical Angles) • What is the measure of Angle 7? (Corresponding Angles) • What is the measure of Angle 6? • What is the measure of Angle 4 (Supplementary Angles)? • What is the measure of Angle 8 (corresponding angles)? 1 2 3 4 5 6 7 8
Alternate Interior Angles • These angles lie between the two lines, on opposite sides of the transversal • Alternate Interior Angles Theorem – If a transversal intersects 2 parallel lines, then alternate interior angles are congruent • Here, angle 3 and angle 6 are Alternate angles. • What is the alternate interior angle that matches up with angle 5? • If angle 5 is 130 degrees, what is the measure of angle 4? • What is the measure of Angle 6? • What is the measure of Angle 3? 1 2 3 4 5 6 7 8
Consecutive Interior Angles, or Same Side Interior Angles • If two angles lie between the two lines on the same side of the transversal they are consecutive interior angles, or same-side interior angles • Same-Side Interior Angles Theorem – If a transversal intersects 2 parallel lines, then same-side interior angles are supplementary • Angle 3 and Angle 5 are Same Side Interior Angles • If angle 5 is 120 degrees, what is the measure of angle 3? • What is the measure of angle 4? • What is the measure of angle 6? 1 2 3 4 5 6 7 8
Alternate Exterior Angles • If two angles lie outside the two lines, on opposite sides of the transversal, then they are Alternate Exterior Angles • Alternate Exterior Angles Theorem – If a transversal intersects 2 parallel lines, then Alternate Exterior Angles are congruent • Angle 1 and Angle 8 are alternate exterior angles • What is the alternate exterior angle for angle 7? • If angle 8 measures 140 degrees, what is the measure of angle 1? 1 2 3 4 5 6 7 8
Check for Learning • Do problems 21 -24 on page 144
Perpendicular Transversals • If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other parallel line • Here, we are given enough information to conclude that line c is perpendicular to line a, because it forms a right (90 degree) angle • Because line a is parallel to line b, we can conclude that line c is also perpendicular to line b a b c
Example • Find the value of X • These are like puzzles: – Here, we see that angle 4 is vertical with an angle that is 125 degrees. Therefore, angle 4 is 125 degrees – We see that angle 4, and the angle (x+15) are same-side interior angles – We know that Same-Side interior angles are supplementary (they add up to 180 degrees) – Therefore 125 + X + 15 = 180 – Or: 140 + X = 180 – Or: X = 40 125 2 3 4 5 X+ 15 7 8
Assignment • Pages 153 -154 Problems 7 -20 (Guided practice) • Worksheets: – 3 -1 – Angle Pairs on two transversals – Transversal on Parallel Lines
Unit 3 Quiz 1 Use the picture at the right to answer the following questions 1. 2. 3. 4. 5. 6. 7. Name a pair of Corresponding Angles Name a pair of Alternate Interior Angles Name a pair of Same Side Interior Angles Name a pair of Alternate Exterior Angles (Yes/No) Are Corresponding Angles congruent? (Yesn/No) are Same Side Interior Angles congruent? If angle 1 measures 130 degrees, what is the measure of angle 8? 8. If angle 2 measures 65 degrees, and angle 6 is (2 x 15), what is x? 9. If angle 7 measures 48 degrees, what is the measure of angle 4? 10. If angle 3 measures (4 x+8) and angle 6 measures (8 x), what is x? 1 2 3 4 5 6 7 8
Unit 3 Final Extra Credit • The measure of Angle 1 is 6 x • The measure of Angle 2 is 2 x-20 1. What is the value of x? 2. How many degrees is the measure of angle 6? 3. What is the measure of angle 5? 4. What type of angle pair are angle 3 and angle 6? 5. What can you conclude about line a and line b? 2 points each, show all work (equations as required) a b 1 2 3 4 5 6 7 8
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