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Do Now

Do Now. Put your homework on your desk ready to be stamped. Take a protractor from the front. What is the relationship between the measure of an angle and the reflex measure of an angle? Using a protractor, draw a clock face with hands to show 8:00.

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Do Now

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  1. Do Now • Put your homework on your desk ready to be stamped. • Take a protractor from the front. • What is the relationship between the measure of an angle and the reflex measure of an angle? • Using a protractor, draw a clock face with hands to show 8:00. • Measure the angle formed by your clock hands in #2 and use correct notation to write the measure of the angle. • Draw a figure that contains at least three angles and includes an angle bisector. Mark any congruent angles.

  2. Angle Relationships

  3. Today’s Objectives • Discover relationships between pairs of angles • Perform investigations and write conjectures • Use conjectures to find missing angles measures. • Use problem solving skills

  4. Using Inductive Reasoning to Discover Geometry • “Discovery consists of looking at the same thing as everyone else and thinking something different.” –Albert Szent-Gyorgyi

  5. Investigation: The Linear Pair Conjecture • Step 1: Draw EHand place a point F between E and H. Choose another point G not on EHand draw FG. You have just created a linear pair of angles. Place the zero edge of your protractor along EH. What do you notice about the sum of the measures of the linear pair of angles? • Step 2: Compare your results with those of your group. Does everyone make the same observation?

  6. Linear Pair Conjecture • If two angles form a linear pair, then ______________________. • the measures of the angles add up to 180o.

  7. Definitions • Supplementary Angles: A pair of angles whose angle measures add up to 180o. • Complementary Angles: A pair of angles whose angle measures add up to 90o.

  8. Investigation:Vertical Angles Conjecture • Step 1: Draw two intersecting lines onto tracing paper. Label the angles as shown. Which angles are vertical angles? • Step 2: Fold the paper so that the vertical angles lie over each other. What do you notice about their measures? • Step 3: Fold the paper so that the other pair of vertical angles lie over each other. What do you notice about their measures? • Step 4: Compare your results with the results of others.

  9. Vertical Angles Conjecture • If two angles are vertical angles, then ____________________. • they have equal measures (they are congruent).

  10. Inductive or Deductive? • To find The Linear Pair Conjecture? • Inductive Reasoning • To find the Vertical Angles Conjecture? • Inductive Reasoning

  11. Using Deductive Reasoning • If we know that The Linear Pair Conjecture is true, can we use deductive reasoning to show that the Vertical Angles Conjecture must be true? • What is the sum of the measures of angles 1 and 2? • 180o • What is the sum of the measures of angles 2 and 3? • 180o • So what can you conclude about the measures of angles 1 and 3. Explain your reasoning.

  12. Is the converse true? • Vertical Angles Conjecture: If two angles are vertical angles, then they have equal measures. • Converse: If two angles have equal measures, then they are vertical angles. • Is this valid? • No. • Counterexample?

  13. Practice • Find the measure of each lettered angle.

  14. Practice • If a=51 and e=52, find the measure of b, c, d, and f.

  15. Practice

  16. Today’s Objectives • Discover relationships between pairs of angles • Perform investigations and write conjectures • Use conjectures to find missing angles measures. • Use problem solving skills

  17. Exit Slip • Use your two conjectures to find the missing angle measures. • Explain how you found each angle measure. • Include which conjecture(s) you used.

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