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Whiteboards Determine whether each statement is true or false. If false, give a counterexample.

Whiteboards Determine whether each statement is true or false. If false, give a counterexample. 1. It two angles are complementary, then they are not congruent. 2. If two angles are congruent to the same angle, then they are congruent to each other. 3. Supplementary angles are congruent.

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Whiteboards Determine whether each statement is true or false. If false, give a counterexample.

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  1. Whiteboards Determine whether each statement is true or false. If false, give a counterexample. 1.It two angles are complementary, then they are not congruent. 2. If two angles are congruent to the same angle, then they are congruent to each other. 3. Supplementary angles are congruent. false; 45° and 45° true false; 60° and 120°

  2. Math Joke of the Day What do you have to know to get top grades in geometry? All the angles!

  3. Remember! Numbers are equal (=) and figures are congruent ().

  4. EXAMPLE 1 Identify the property that justifies each statement. A. QRS  QRS B. m1 = m2 so m2 = m1 C. AB  CD and CD  EF, so AB EF. D. 32° = 32°

  5. WHITEBOARDS Identify the property that justifies each statement. • x = y and y = z, so x = z. • DEF  DEF • ABCD, so CDAB.

  6. Get to Know Your Group Members! Directions: Solve the given problem with your group members. Make sure everyone understands how to do it because I will pick one person at random to present. A, B, C, D, and Eare on line l such that B is the midpoint of AE, C is the midpoint of BE, and D is the midpoint of CE. If AD = 10, then what is AE?

  7. 2.6: Geometric Proof • Learning Objective • SWBAT write two-column proofs

  8. When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them.

  9. Helpful Hint When a justification is based on more than the previous step, you can note this after the reason, as in Example 1 Step 3.

  10. Write a justification for each step, given that B is the midpoint of AC and ABEF. 1. Bis the midpoint of AC. 2. AB BC 3. AB EF 4. BC EF Example 1 Given information Def. of mdpt. Given information Trans. Prop. of 

  11. A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column.

  12. Example 2: Completing a Two-Column Proof Fill in the blanks to complete the two-column proof. Given: XY Prove: XY  XY Reflex. Prop. of =

  13. Example 3: Fill in the blanks to complete a two-column proof of one case of the Congruent Supplements Theorem. Given: 1 and 2 are supplementary, and 2 and 3 are supplementary. Prove: 1  3 Proof: • 1 and 2 are supp., and 2 and 3 are supp. b. m1 + m2 = m2 + m3 c. Subtr. Prop. of = d. 1  3

  14. Example 4: With your Table Use the given plan to write a two-column proof if one case of Congruent Complements Theorem. Given: 1 and 2 are complementary, and 2 and 3 are complementary. Prove: 1  3 Plan: The measures of complementary angles add to 90° by definition. Use substitution to show that the sums of both pairs are equal. Use the Subtraction Property and the definition of congruent angles to conclude that 1 3

  15. Example 4 Continued Given 1 and 2 are complementary. 2 and 3 are complementary. m1 + m2 = 90° m2 + m3 = 90° Def. of comp. s m1 + m2 = m2 + m3 Subst. Reflex. Prop. of = m2 = m2 m1 = m3 Subtr. Prop. of = 1  3 Def. of  s

  16. Example 5: Write a proof, given that A and Bare supplementary and mA = 45°. 1. A and Bare supplementary. mA = 45° Given information Def. of supp s 2. mA + mB= 180° Subst. Prop of = 3. 45° + mB= 180° Steps 1, 2 Subtr. Prop of = 4. mB= 135°

  17. Example 6: Use the given plan to write a two-column proof. Given: 1 and 2 are supplementary, and 1  3 Prove: 3 and 2 are supplementary. Plan: Use the definitions of supplementary and congruent angles and substitution to show that m3 + m2 = 180°. By the definition of supplementary angles, 3 and 2 are supplementary.

  18. Example 6 Continued Given 1 and 2 are supplementary. 1  3 m1 + m2 = 180° Def. of supp. s m1 = m3 Def. of s Subst. m3 + m2 = 180° Def. of supp. s 3 and 2 are supplementary

  19. Exit Ticket Write a justification for each step, given that mABC = 90° and m1 = 4m2. 1. mABC = 90° and m1 = 4m2 2. m1 + m2 = mABC 3. 4m2 + m2 = 90° 4. 5m2= 90° 5. m2= 18° Given  Add. Post. Subst. Simplify Div. Prop. of =.

  20. VOCABULARY LOGS With your group, predict what words/ postulates/ theorems should go into your vocabulary log. Your vocab log is due September 25th- the day of your unit test.

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