Lesson 7.6 Parallelograms pp. 291-295

Objectives: 1. To prove the SAS Congruence Theorem for parallelograms. 2. To identify and prove the basic properties of parallelograms.

Theorem 7.15 The opposite sides of a parallelogram are congruent.

Theorem 7.15 The opposite sides of a parallelogram are congruent.

Theorem 7.16 SAS Congruence for Parallelograms. A B C D P Q R S

Theorem 7.17 A quadrilateral is a parallelogram if and only if the diagonals bisect one another.

Theorem 7.18 Diagonals of a rectangle are congruent.

Theorem 7.19 The sum of the measures of the four angles of every convex quadrilateral is 360°. B A C D

Theorem 7.20 Opposite angles of a parallelogram are congruent.

Theorem 7.21 Consecutive angles of a parallelogram are supplementary. 2 1

Theorem 7.22 If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 7.23 A quadrilateral with one pair of parallel sides that are congruent is a parallelogram.

Practice: Determine whether the given figure must be a parallelogram Practice: Determine whether the given figure must be a parallelogram. Be ready to explain your answer. 1. Yes 2. No

Practice: Determine whether the given figure must be a parallelogram Practice: Determine whether the given figure must be a parallelogram. Be ready to explain your answer. 1. Yes 2. No A B D C 68° 112°

Practice: Determine whether the given figure must be a parallelogram Practice: Determine whether the given figure must be a parallelogram. Be ready to explain your answer. 1. Yes 2. No

Practice: Determine whether the given figure must be a parallelogram Practice: Determine whether the given figure must be a parallelogram. Be ready to explain your answer. 1. Yes 2. No

Practice: Determine whether the given figure must be a parallelogram Practice: Determine whether the given figure must be a parallelogram. Be ready to explain your answer. 1. Yes 2. No 100° 80°

Practice: Determine whether the given figure must be a parallelogram Practice: Determine whether the given figure must be a parallelogram. Be ready to explain your answer. 1. Yes 2. No 120° 60°

Homework pp. 293-295

►A. Exercises Using parallelogram ABCD, find the measures of the indicated angles. 1. C A D B C 48º 55º

►A. Exercises Using parallelogram ABCD, find the measures of the indicated angles. 2. ADC A D B C 48º 55º

►A. Exercises Using parallelogram ABCD, find the measures of the indicated angles. 3. ABD A D B C 48º 55º

►A. Exercises Using parallelogram ABCD, find the measures of the indicated angles. 4. The four angles of the parallelogram combined A D B C 48º 55º

►A. Exercises Using parallelogram ABCD, find the measures of the indicated angles. 5. An exterior angle of the parallelogram at angle C A D B C 48º 55º

►A. Exercises 6. Adjacent angles form a linear pair. Disprove the following statements. Remember that you disprove a statement by proving that it is false. This can be done with an illustration or a counterexample. 6. Adjacent angles form a linear pair.

►A. Exercises 7. Alternate interior angles are congruent. Disprove the following statements. Remember that you disprove a statement by proving that it is false. This can be done with an illustration or a counterexample. 7. Alternate interior angles are congruent.

►A. Exercises Disprove the following statements. Remember that you disprove a statement by proving that it is false. This can be done with an illustration or a counterexample. 8. Every pair of supplementary angles form a linear pair.

►A. Exercises 9. The acute angles of a triangle are complementary. Disprove the following statements. Remember that you disprove a statement by proving that it is false. This can be done with an illustration or a counterexample. 9. The acute angles of a triangle are complementary.

►A. Exercises Disprove the following statements. Remember that you disprove a statement by proving that it is false. This can be done with an illustration or a counterexample. 10. If two triangles have a pair of congruent angles, then the other pairs of angles are congruent.

►B. Exercises 12. Given: ABCD with diagonals AC and BD bisecting each other at E Prove: ABCD is a parallelogram A D B C E

►B. Exercises 12. Given: ABCD with diagonals AC and BD bisecting each other at E Prove: ABCD is a parallelogram A D B C E

►B. Exercises 14. Given: Convex quadrilateral ABCD Prove: mABC + mBCD + mCDA + mDAB = 360º C D B A

►B. Exercises 16. Given: ABCD is a parallelogram Prove: A & B are supplementary C B 3 1 2 D A

■ Cumulative Review Suppose two segments must be proved congruent. What reason could you use that involves the concept named? 25. distances

■ Cumulative Review Suppose two segments must be proved congruent. What reason could you use that involves the concept named? 26. bisectors

■ Cumulative Review Suppose two segments must be proved congruent. What reason could you use that involves the concept named? 27. one triangle

■ Cumulative Review Suppose two segments must be proved congruent. What reason could you use that involves the concept named? 28. two triangles

■ Cumulative Review Suppose two segments must be proved congruent. What reason could you use that involves the concept named? 29. a circle

■ Cumulative Review Suppose two segments must be proved congruent. What reason could you use that involves the concept named? 30. a parallelogram

Analytic Geometry Midpoints

If M is the midpoint of AB where A(x1, y1) and B(x2, y2), Midpoint Formula If M is the midpoint of AB where A(x1, y1) and B(x2, y2), ÷ ø ö ç è æ + = 2 y x then 1 , M

Find the coordinate of the midpoint between the points (3, -5) and (1, -6). ÷ ø ö ç è æ - + = 2 6 5 1 3 M ) ( , ÷ ø ö ç è æ - = 2 11 ,

Find the coordinate of the midpoint between the two points. 1. (4, 8) and (2, -3)

Find the coordinate of the midpoint between the two points. 2. (3, 5) and (3, 9)

Find the coordinate of the midpoint between the two points. 3. (-1, -4) and (6, -2)