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Chapter 6 Lesson 3

Chapter 6 Lesson 3. Proving that a Quadrilateral is a Parallelogram. Warm-up. Find the coordinate of the midpoints of each line segment. What is the relationship between AC and BD? 2) Find the slopes of each line. 3) Are AB and DC parallel? How do you know?

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Chapter 6 Lesson 3

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  1. Chapter 6 Lesson 3 Proving that a Quadrilateral is a Parallelogram

  2. Warm-up Find the coordinate of the midpoints of each line segment. What is the relationship between AC and BD? 2) Find the slopes of each line. 3) Are AB and DC parallel? How do you know? 4) What type of figure is ABCD? D B C A

  3. Writing Proofs Given <L ≅ <ENS Opposite angles are Congruent in Parallelograms <ENS≅<HNG Vertical angles are Congruent <T ≅ <HNG Opposite angles are Congruent in Parallelograms <L ≅ <T Transitive Property of Congruence

  4. Today’s Objective • Prove that a quadrilateral is a parallelogram

  5. Evidence that we can use for Proof • Theorem 6-5: If the diagonal of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. • Theorem 6-6: if one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. • Theorem 6-7: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. • Theorem 6-8: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

  6. Proof Box • You can say you have a quadrilateral if you have: • Diagonals that bisect each other • Two opposite sides that are congruent and parallel • Two pairs of opposite sides that are congruent • Two pairs of opposite angles that are congruent I think it looks like a parallelogram is not an option!

  7. Example 1: Finding Values for Parallelograms Find the values for x and y to make MLPN a parallelogram. Set 3x = x + 5 Solve: 2x = 5 x = 2.5 2) Set 2y – 7 = y + 2 Solve: y = 9

  8. Example 2: Is it a Parallelogram? • Can you conclude this is a parallelogram? Explain. Yes, both pairs of opposite angles are congruent. No, adjacent sides are congruent. This could be a kite.

  9. Example 3: Real-World Connection A parallel rule is a navigation tool used to plot ship routes on charts. It is made of Two rulers connected with congruent crossbars such that AD = BC and AB = DC. You place one ruler on the line connecting the ship’s present position to its destination. Then you move the other ruler onto the chart’s compass to find the direction of route. Explain why this instrument works.

  10. Answer • The crossbars and the sections of the rulers are congruent no matter how they are positioned, so ABCD is always a parallelogram. • Therefore, the rulers are parallel. • The direction the ship should travel is the same as the direction shown on the chart’s compass.

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