8.3 Tests for Parallelograms

Objectives Recognize the conditions that ensure a quadrilateral is a parallelogram. Prove that a set of points forms a parallelogram in the coordinate plane.

Conditions for a Parallelogram Obviously, if the opposite sides of a quadrilateral are parallel, then it is a parallelogram; but there are other tests we can also apply to a quadrilateral to test whether it is a parallelogram or not.

Conditions for a Theorems Theorem 8.9 – If both pairs of opposite sides are ≅, then the quad. is a . Theorem 8.10 – If both pairs of opposite s are ≅, then the quad. is a . Theorem 8.11 – If diagonals bisect each other, then the quad. is . Theorem 8.12 – If one pair of opposite sides is ║ and ≅, then the quad. is a .

Example 1: Write a paragraph proof of the statement: If a diagonal of a quadrilateral divides the quadrilateral into two congruent triangles, then the quadrilateral is a parallelogram. Given: Prove: ABCD is a parallelogram. Proof: CPCTC. By Theorem 8.9, if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Therefore, ABCD is a parallelogram.

Your Turn: Write a paragraph proof of the statement: If two diagonals of a quadrilateral divide the quadrilateral into four triangles where opposite triangles are congruent, then the quadrilateral is a parallelogram. Given: Prove: WXYZ is a parallelogram.

Your Turn: Proof: by CPCTC. By Theorem 8.9, if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Therefore, WXYZ is a parallelogram.

Example 2: Some of the shapes in this Bavarian crest appear to be parallelograms. Describe the information needed to determine whether the shapes are parallelograms. Answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is a congruent and parallel, the quadrilateral is a parallelogram. If both pairs of opposite angles are congruent or if the diagonals bisect each other, the quadrilateral is a parallelogram.

Your Turn: The shapes in the vest pictured here appear to be parallelograms. Describe the information needed to determine whether the shapes are parallelograms. Answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and parallel, the quadrilateral is a parallelogram. If both pairs of opposite angles are congruent or if the diagonals bisect each other, the quadrilateral is a parallelogram.

Example 3: Determine whether the quadrilateral is a parallelogram. Justify your answer. Answer: Each pair of opposite sides have the same measure. Therefore, they are congruent. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

Your Turn: Determine whether the quadrilateral is a parallelogram. Justify your answer. Answer: One pair of opposite sides is parallel and has the same measure, which means these sides are congruent. If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

Tests for Parallelograms Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. Both pairs of opposite angles are congruent. The diagonals bisect each other. A pair of opposite sides is both parallel and congruent.

Example 4a: Find x so that the quadrilateral is a parallelogram. A B C Opposite sides of a parallelogram are congruent.

Example 4a: Substitution Distributive Property Subtract 3x from each side. Add 1 to each side. Answer: When x is 7, ABCD is a parallelogram.

Example 4b: Find y so that the quadrilateral is a parallelogram. F D E Opposite angles of a parallelogram are congruent.

Example 4b: Substitution Subtract 6y from each side. Subtract 28 from each side. Divide each side by –1. Answer: DEFG is a parallelogram when y is 14.

Your Turn: Find m and n so that each quadrilateral is a parallelogram. b. Answer: Answer:

Parallelograms on the Coordinate Plane We can use the Distance Formula and the Slope Formula to determine if a quadrilateral is a parallelogram on the coordinate plane. Just pick one of the tests… and apply either or both of the formulas.

Example 5a: COORDINATE GEOMETRY Determine whether the figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and D(1, –1) is a parallelogram. Use the Slope Formula.

Example 5a: If the opposite sides of a quadrilateral are parallel, then it is a parallelogram. Answer: Since opposite sides have the same slope, Therefore, ABCD is a parallelogram by definition.

Example 5b: COORDINATE GEOMETRY Determine whether the figure with vertices P(–3, –1), Q(–1, 3), R(3, 1), and S(1, –3) is a parallelogram. Use the Distance and Slope Formulas.

Example 5b: First use the Distance Formula to determine whether the opposite sides are congruent.

Example 5b: Next, use the Slope Formula to determine whether and have the same slope, so they are parallel. Answer: Since one pair of opposite sides is congruent and parallel, PQRS is a parallelogram.

Your Turn: Determine whether the figure with the given vertices is a parallelogram. Use the method indicated. a. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1); Slope Formula

Your Turn: Answer: The slopes of and the slopes of Therefore, Since opposite sides are parallel, ABCD is a parallelogram.

Your Turn: Determine whether the figure with the given vertices is a parallelogram. Use the method indicated. Distance and Slope Formulas b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2);

Your Turn: Answer: Since the slopes of Since one pair of opposite sides is congruent and parallel, LMNO is a parallelogram.

Assignment Pre-AP Geometry: Pg. 421 #13 – 32 25 – 28