7.3 Parallelogram Proofs
Theorem: If both pair of
opposite angles of a quadrilateral are congruent, the quadrilateral is a
parallelogram.
Proof
Given: ÐP @ ÐR, ÐRQ P@ ÐRSP
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, mÐP + mÐQ + mÐR + mÐS = 360.
Since
ÐP @ ÐR and ÐQ @ ÐS, mÐP = mÐR and mÐQ = mÐS. Substitute to find that mÐP + mÐP + mÐQ + mÐQ = 360,
or 2(mÐP) + 2(mÐQ) =
360. Dividing each side of the equation by 2 yields mÐP + mÐQ = 180.
This means that consecutive angles are supplementary and _{}║_{}.
Likewise, 2mÐP + 2mÐS = 360, or mÐP + mÐS = 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.