Showing Quadrilaterals are Parallelograms Section 6.3

Objectives Show that a quadrilateral is a parallelogram.

Key Vocabulary None

Theorems 6.6 – 6.9 Conditions for Parallelograms

Conditions for a Parallelogram Obviously, if the opposite sides of a quadrilateral are parallel (the definition of a parallelogram), 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.

Recall if a quadrilateral was a parallelogram then Both pairs of opposite sides are congruent Both pairs of opposite angles are congruent The diagonals bisect each other One angle is supplementary to both consecutive angles x˚x˚ x˚x˚ y˚y˚ x ˚ +y ˚ =180 ˚

The Converses are also true; A quadrilateral is a Parallelogram if: Both pairs of opposite sides are equal… Both pairs of opposite angles are equal… The diagonals bisect each other… One angle is supplementary to both consecutive angles… x˚x˚ x˚x˚ y˚y˚ x ˚ +y ˚ =180 ˚

Theorems Conditions for Parallelograms

Investigation In this lesson, we will find ways to show that a quadrilateral is a parallelogram. Obviously, if the opposite sides are parallel, then the quadrilateral is a parallelogram. But could we use other properties besides the definition to see if a shape is a parallelogram?

Property 1 We know that the opposite sides of a parallelogram are congruent. What about the converse? If we had a quadrilateral whose opposite sides are congruent, then is it also a parallelogram? Step 1: Draw a quadrilateral with congruent opposite sides.

Property 1 Step 2: Draw diagonal AD. Notice this creates two triangles. What kind of triangles are they? by SSS 

Property 1 Step 3: Since the two triangles are congruent, what must be true about  BDA and  CAD? by CPCTC

Property 1 Step 4: Now consider AD to be a transversal. What must be true about BD and AC? by Converse of Alternate Interior Angles Theorem

Property 1 Step 5: By a similar argument, what must be true about AB and CD? by Converse of Alternate Interior Angles Theorem

Property 1 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 6.6 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

Property 2 We know that the opposite angles of a parallelogram are congruent. What about the converse? If we had a quadrilateral whose opposite angles are congruent, then is it also a parallelogram? Step 1: Draw a quadrilateral with congruent opposite angles.

Property 2 Step 2: Now assign the congruent angles variables x and y. What is the sum of all the angles? What is the sum of x and y ?

Property 2 Step 3: Consider AB to be a transversal. Since x and y are supplementary, what must be true about BD and AC? by Converse of Consecutive Interior Angles Theorem

Property 2 Step 4: By a similar argument, what must be true about AB and CD? by Converse of Consecutive Interior Angles Theorem

Property 2 If both pairs of opposite angles 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. ABCD is a parallelogram.

Property 3 This property is not a converse. The question is, if we had a quadrilateral with one angle supplementary to both of it’s consecutive angles, then is it also a parallelogram? Step 1: Draw a quadrilateral with one angle ( ∠ B) is supplementary to both its consecutive angles ( ∠ A & ∠ C). A BC D y˚y˚ x˚x˚ x˚x˚ x ˚ +y ˚ =180 ˚

Property 3 Step 2: By the congruent supplements theorem 2.2, if two angles are supplementary to the same angle then they are congruent. Thus, ∠ A ≅∠ B. A BC D y˚y˚ x˚x˚ x˚x˚ x ˚ +y ˚ =180 ˚

Property 3 Step 3: By the quadrilateral interior angles theorem, m ∠ A+m ∠ B+m ∠ C+m ∠ D=360˚. Since, ∠ A and ∠ B are supplementary, then m ∠ A+m ∠ B=180˚. By substitution 180 ˚ +m ∠ C+m ∠ D=360 ˚ or m ∠ C+m ∠ D=180 ˚. Then, ∠ C and ∠ D are supplementary. A BC D y˚y˚ x˚x˚ x˚x˚ x ˚ +y ˚ =180 ˚

Property 3 Step 4: Since ∠ B is supplementary to ∠ C and ∠ D is supplementary to ∠ C also, then ∠ B ≅∠ D by the congruent supplements theorem. A BC D y˚y˚ x˚x˚ x˚x˚ x ˚ +y ˚ =180 ˚ y˚y˚

Property 3 Step 5: Finally, since the opposite angles of our quadrilateral are congruent, what must be true about our quadrilateral? A BC D

Property 3 If one angle of a quadrilateral is supplementary to both of it’s consecutive angles, then the quadrilateral is a parallelogram. A BC D y˚y˚ x˚x˚ x˚x˚ x ˚ +y ˚ =180 ˚

Theorem 6.8 If one angle of a quadrilateral is supplementary to both of it’s consecutive angles, then the quadrilateral is a parallelogram. ABCD is a parallelogram. y˚y˚ x˚x˚ x˚x˚ x ˚ +y ˚ =180 ˚ A BC D

Property 4 We know that the diagonals of a parallelogram bisect each other. What about the converse? If we had a quadrilateral whose diagonals bisect each other, then is it also a parallelogram? Step 1: Draw a quadrilateral with diagonals that bisect each other.

Property 4 Step 2: What kind of angles are  BEA and  CED? So what must be true about them? by Vertical Angles Congruence Theorem

Property 4 Step 3: Now what must be true about AB and CD? by SAS  and CPCTC

Property 4 Step 4: By a similar argument, what must be true about BD and AC? by SAS  and CPCTC

Property 4 Step 5: Finally, if the opposite sides of our quadrilateral are congruent, what must be true about our quadrilateral? ABDC is a parallelogram by Theorem 6.9

Property 4 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

Summary of Theorems: Conditions for Parallelograms Theorem 6.6 – If both pairs of opposite sides are ≅, then the quad. is a. Theorem 6.7 – If both pairs of opposite  s are ≅, then the quad. is a. Theorem 6.8 – If one angle is supplementary to both its consecutive angles, then the quad. is a. Theorem 6.9 – If diagonals bisect each other, then the quad. is.

Example 1 Determine whether the quadrilateral is a parallelogram. Justify your answer. Answer:Each pair of opposite sides has 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 A.Both pairs of opp. sides ||. B.Both pairs of opp. sides . C.Both pairs of opp.  ’s . D.One pair of opp. sides both || and . Which method would prove the quadrilateral is a parallelogram?

Example 2 MECHANICS Scissor lifts, like the platform lift shown below, are commonly applied to tools intended to lift heavy items. In the diagram,  A   C and  B   D. Explain why the consecutive angles will always be supplementary, regardless of the height of the platform.

Example 2 Answer:Since both pairs of opposite angles of quadrilateral ABCD are congruent, ABCD is a parallelogram by Theorem Theorem 6.5 states that consecutive angles of parallelograms are supplementary. Therefore, m  A + m  B = 180 and m  C + m  D = 180. By substitution, m  A + m  D = 180 and m  C + m  B = 180.

Your Turn A.  A   B B.  A   C C.AB  BC D.m  A + m  C = 180 The diagram shows a car jack used to raise a car from the ground. In the diagram, AD  BC and AB  DC. Based on this information, which statement will be true, regardless of the height of the car jack.

Example 3 Find x and y so that the quadrilateral is a parallelogram. Opposite sides of a parallelogram are congruent.

Example 3 Substitution Distributive Property Add 1 to each side. Subtract 3x from each side. AB = DC

Example 3 Answer:So, when x = 7 and y = 5, quadrilateral ABCD is a parallelogram. Substitution Distributive Property Add 2 to each side. Subtract 3y from each side.

Your Turn A.m = 2 B.m = 3 C.m = 6 D.m = 8 Find m so that the quadrilateral is a parallelogram.

Tests for Parallelograms 1.Both pairs of opposite sides are parallel. 2.Both pairs of opposite sides are congruent. 3.Both pairs of opposite angles are congruent. 4.The diagonals bisect each other. 5.One angle is supplementary to both its consecutive angles.

PRACTICE

6 6 Determine whether the quadrilateral is a parallelogram.

50° 130° Determine whether the quadrilateral is a parallelogram.

Use Opposite Sides Example 1 Tell whether the quadrilateral is a parallelogram. Explain your reasoning. SOLUTION The quadrilateral is not a parallelogram. It has two pairs of congruent sides, but opposite sides are not congruent.

Use Opposite Angles Example 2 SOLUTION The quadrilateral is a parallelogram because both pairs of opposite angles are congruent. Tell whether the quadrilateral is a parallelogram. Explain your reasoning.

Use Consecutive Angles Example 3 Tell whether the quadrilateral is a parallelogram. Explain your reasoning. a.b.c. SOLUTION  U is supplementary to  T and  V (85° + 95° = 180°). So, by Theorem 6.8, TUVW is a parallelogram. a.  G is supplementary to  F (55° + 125° = 180°), but  G is not supplementary to  H (55° + 120° ≠ 180°). So, EFGH is not a parallelogram. b.

Use Consecutive Angles Example 3  D is supplementary to  C (90° + 90° = 180°), but you are not given any information about  A or  B. Therefore, you cannot conclude that ABCD is a parallelogram. c.

Use Diagonals Example 4 Tell whether the quadrilateral is a parallelogram. Explain your reasoning. b. SOLUTION a.The diagonals of JKLM bisect each other. So, by Theorem 6.9, JKLM is a parallelogram. a. The diagonals of PQRS do not bisect each other. So, PQRS is not a parallelogram. b.

Your Turn: Use Opposite Sides and Opposite Angles In Exercises 1 and 2, tell whether the quadrilateral is a parallelogram. Explain your reasoning. 1. In quadrilateral WXYZ, WX = 15, YZ = 20, XY = 15, and ZW = 20. Is WXYZ a parallelogram? Explain your reasoning ANSWER Yes; both pairs of opposite sides are congruent. No; opposite angles are not congruent. ANSWER No; opposite sides are not congruent.

Your Turn: Use Consecutive Angles and Diagonals Tell whether the quadrilateral is a parallelogram. Explain your reasoning ANSWER No; opposite angles are not congruent (or consecutive angles are not supplementary). Yes; one angle is supplementary to both of its consecutive angles. ANSWER

Your Turn: Use Consecutive Angles and Diagonals ANSWER Yes; the diagonals bisect each other. No; the diagonals do not bisect each other. ANSWER Tell whether the quadrilateral is a parallelogram. Explain your reasoning.

Assignment Pg #9 – 13 odd, 19 – 27 odd, 46 – 49, 51, 53