7.3 Parallelogram Proofs

 

Theorem: If both pair of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.

Proof

Given: P @ R, RQ P@ RSP

Prove: PQRS is a parallelogram

 

Paragraph Proof:

Because two points determine a line, we can draw . We now have two triangles. We know the sum of the angle measures of a triangle is 180, so the sum of the angle measures of two triangles is 360. Therefore, mP + mQ + mR + mS = 360.

 

Since P @ R and Q @ S, mP = mR and mQ = mS. Substitute to find that mP + mP + mQ + mQ = 360, or 2(mP) + 2(mQ) = 360. Dividing each side of the equation by 2 yields mP + mQ = 180. This means that consecutive angles are supplementary and .

 

Likewise, 2mP + 2mS = 360, or mP + mS = 180. These consecutive supplementary angles verify that . Opposite sides are parallel, so ABCD is a parallelogram.

 

 

Example 2 Properties of Parallelograms

CONSTRUCTION Wood lattice panels are

usually made in a configuration of parallelograms.

Explain how the person who made the panels could

verify that the overlapped boards form

parallelograms.

The overlapping boards form a parallelogram when each pair of opposite segments is congruent. If the person making the panels measures each pair of opposite segments, and they are the same, then the boards form parallelograms.

 

 

Example 3 Properties of Parallelograms

Determine whether the quadrilateral is a

parallelogram. Justify your answer.

The measure of the angle adjacent to the exterior 118 angle is 62

since the two form a linear pair. Each pair of opposite angles has

the same measure. Therefore, they are congruent. If both pairs of

opposite angles are congruent, the quadrilateral is a parallelogram.


Example 4 Find Measures

ALGEBRA Find x and y so that each quadrilateral is a parallelogram.

a.

 

 

 

 

Opposite sides of a parallelogram are congruent.

@ Opp. sides of a

parallelogram are @.

AB = DC Def. of @ segs.

5x + 12 = 8x Substitution

12 = 3x Subtract 5x.

4 = x Divide by 3.

@ Opp. sides of a

parallelogram are @.

AD = BC Def. of @ segs.

3y + 6 = 5y - 6 Substitution

12 = 2y Subtract 3y and add 6.

6 = y Divide by 2.

 

So, when x is 4 and y is 6, ABCD is a parallelogram.

 

b.

 

 

 

 

 

Diagonals in a parallelogram bisect each other.

@ Diag. of a parallelogram

bisect each other.

MR = PR Def. of @ segs.

4x = 5x - 4 Substitution

-x = -4 Subtract 5x.

x = 4 Divide by -1.

@ Diag. of a parallelogram

bisect each other.

QR = NR Def. of @ segs.

2y = y + 7 Substitution

y = 7 Subtract y.

 

 

MNPQ is a parallelogram when x = 4 and y = 7.

 

 

Example 5 Use Slope and Distance

COORDINATE GEOMETRY Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.

a. A(0, -4), B(-6, 1), C(3, 7), D(9, 2); Slope Formula

If the opposite sides of a quadrilateral are parallel,
then it is a parallelogram.

slope of = or - slope of = or -

slope of = or slope of = or

 

Since opposite sides have the same slope, and
. Therefore, ABCD is a parallelogram by definition.


b. W(3, 1), X(-1, -4), Y(-2, 2), Z(2, 7); Distance and Slope Formulas

First use the Distance Formula to determine whether
the opposite sides are congruent.

WX =

= or

ZY =

= or

 

Since WX = ZY, @.

 

Next, use the Slope Formula to determine whether .

slope of = or slope of = or

and have the same slope, so they are parallel. Since one pair of opposite sides is congruent and parallel, WXYZ is a parallelogram.