Find the value of each variable in the parallelogram. A parallelogram with one pair of opposite angle labeled left-parenthesis b minus 10 right-parenthesis degrees and c degrees, and the other pair of opposite angles are labeled d degrees and left-parenthesis b plus 10 right-parenthesis degrees. $b=$ $c=$ $d=$
The Correct Answer and Explanation is :
In a parallelogram, opposite angles are always congruent, meaning that the measure of one angle equals the measure of the opposite angle. This property is key to solving for the values of the variables in the given problem.
We are given the following angle labels in the parallelogram:
- One pair of opposite angles are ( (b – 10)^\circ ) and ( c^\circ ).
- The other pair of opposite angles are ( d^\circ ) and ( (b + 10)^\circ ).
Step 1: Use the property of opposite angles
Since opposite angles are congruent, we can set up the following equations based on the given angle relationships:
- ( b – 10 = c ) (for the first pair of opposite angles)
- ( b + 10 = d ) (for the second pair of opposite angles)
Step 2: Use the sum of angles in a parallelogram
In a parallelogram, consecutive angles are supplementary, meaning the sum of the measures of two adjacent angles is ( 180^\circ ). This gives us two more relationships:
- The sum of the angles ( (b – 10)^\circ ) and ( (b + 10)^\circ ) must equal ( 180^\circ ): [
(b – 10) + (b + 10) = 180
] Simplifying the equation:
[
2b = 180
]
[
b = 90
]
Step 3: Solve for the other variables
Now that we know ( b = 90^\circ ), we can substitute this value into the earlier equations to find ( c ) and ( d ).
From ( b – 10 = c ):
[
90 – 10 = c
]
[
c = 80^\circ
]
From ( b + 10 = d ):
[
90 + 10 = d
]
[
d = 100^\circ
]
Final Answer:
Thus, the values of the variables are:
- ( b = 90^\circ )
- ( c = 80^\circ )
- ( d = 100^\circ )
Explanation:
The key to solving this problem is recognizing the properties of parallelograms, namely that opposite angles are equal, and adjacent angles are supplementary. By setting up equations based on these properties and solving them step-by-step, we can determine the values of all the unknown angles in the parallelogram.