Theorem A: Opposite angles of a parallelogram are congruent.
Theorem B: Consecutive angles are supplementary (their sum is 180°).
The sum of all the angles of a parallelogram is 360°.
Theorem C: Opposite sides of a parallelogram are parallel and congruent.
1) ∠ 1 = 78° Its opposite angle = 78° (Theorem A)
Measure each angle of the other pair of opposite angles (Theorem B):
= 180 - 78
= 102°
ANSWER: If a given angle measures 78°, the other angles measure 78°, 102°, and 102°, respectively.
2) Consecutive angles:
∠1: x ∠2: x + 72
By Theorem B:
(x) + (x+72) = 180
2x + 72 = 180
2x/2 = (180-72)/2
x = 54
The consecutive angles are:
∠1 = x = 54° Its opposite angle = 54°
∠2 = 54+72 = 126° It opposite angles = 126
Check: The difference between consecutive angles is 72.
126 - 54 = 72
72 = 72 (True)
ANSWER: The measures of all the angles are 54°, 54°, 126°, and 126°.
3) Consecutive angles:
∠1 = 4x ∠2 = 5x
Theorem B:
4x + 5x = 180
9x/9 = 180/9
x = 20
The angles' measures:
∠1: 4x = 4(20) = 80° Its opposite angle = 80°
∠2:5x = 5(20) = 100° Its opposite angle = 100°
ANSWER: The measures of each angle are 80°, 100°, 80°, and 100°,
Check:
The sum of the measure of the angles of a parallelogram is 360°.
2 (80°) + 2(100°) = 360°
160° + 200° = 360°
360° = 360° (True)
4) Theorem: Opposite sides of a parallelogram are parallel and congruent.
AB: x
BC: 2x
Perimeter: 90 cm
Perimeter of Parallelogram = 2 (x) + 2(2x)
90 = 2x + 4x
90/6 = 6x/6
x = 15
Dimensions:
Width: AB = x = 15 cm Opposite side = 15 cm
Length: BC = 2x = 2(15) = 30 cm Opposite side = 30 cm
ANSWER: The measures of the sides are 15 cm, 30 cm, 15 cm, and 30 cm.
Check:
2(15) + 2(30) = 90
30 + 60 = 90
90 = 90 (True)
