Download 6-3 Proving That a Quadrilateral is a Parallelogram

Proving That a Quadrilateral is a Parallelogram
LESSON 6-3
Additional Examples
Find values of x and y for which ABCD must be a
parallelogram.
If the diagonals of quadrilateral ABCD bisect each
other, then ABCD is a parallelogram by Theorem 6-5.
Write and solve two equations to find values of x and y
for which the diagonals bisect each other.
10x – 24 = 8x + 12
2x – 24 = 12
Diagonals of parallelograms
bisect each other.
Collect the variable terms
on one side.
2x = 36
Solve.
x = 18
If x = 18 and y = 89, then ABCD is a parallelogram.
HELP
2y – 80 = y + 9
y – 80 = 9
y = 89
Quick Check
GEOMETRY
Proving That a Quadrilateral is a Parallelogram
LESSON 6-3
Additional Examples
a.
Determine whether the quadrilateral is a parallelogram. Explain.
a. All you know about the quadrilateral is that
only one pair of opposite sides is
congruent.
Therefore, you cannot conclude that the
quadrilateral is a parallelogram.
b. The sum of the measures of the angles of a
polygon is (n – 2)180, where n represents the
number of sides, so the sum of the measures
of the angles of a quadrilateral is
(4 – 2)180 = 360.
If x represents the measure of the unmarked
angle, x + 75 + 105 + 75 = 360, so x = 105.
Because both pairs of opposite angles are congruent, the quadrilateral is a
parallelogram by Theorem 6-6.
Quick Check
b.
HELP
GEOMETRY
Proving That a Quadrilateral is a Parallelogram
LESSON 6-3
Additional Examples
The captain of a fishing boat plots a course toward a
school of bluefish. One side of a parallel rule connects the boat
with the school of bluefish. The other side makes a 36° angle
north of due east on the chart’s compass. Explain how the
captain knows in which direction to sail to reach the bluefish.
Because both sections of the rulers and the crossbars are congruent, the
rulers and crossbars form a parallelogram.
Therefore, the angle shown on the chart’s compass is congruent to the
angle the boat should travel, which is 36° north of due east.
Quick Check
HELP
GEOMETRY