In the diagram of parallelogram ABCD, m∠A = 2x - 10 and m∠B = 5x + 15. Find x.

To find the value of x, we can use the properties of a parallelogram. In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (the sum of their measures is 180 degrees).

Since ABCD is a parallelogram, ∠A and ∠B are consecutive angles. Thus, the sum of m∠A and m∠B is 180 degrees.

Given m∠A = 2x - 10 and m∠B = 5x + 15, we can write the equation:

m∠A + m∠B = 180

Substitute the given angle measures into the equation:

(2x - 10) + (5x + 15) = 180

Now combine like terms (2x and 5x; -10 and 15):

2x + 5x + (-10 + 15) = 180 7x + 5 = 180

Next, subtract 5 from both sides to isolate the term with x:

7x + 5 - 5 = 180 - 5 7x = 175

Finally, divide both sides by 7 to solve for x:

7x / 7 = 175 / 7 x = 25

Therefore, the value of x is 25.

Extra: The properties of a parallelogram are essential for solving this problem. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Due to this parallel nature, a parallelogram also has the following properties:

1. Opposite sides are equal in length. 2. Opposite angles are equal (m∠A = m∠C and m∠B = m∠D). 3. Consecutive angles are supplementary (m∠A + m∠B = 180 degrees and m∠C + m∠D = 180 degrees). 4. The diagonals of a parallelogram bisect each other.

Understanding these properties allows students to solve various geometric problems involving parallelograms, including finding missing angle measures and side lengths. In our problem, we used the fact that consecutive angles in a parallelogram are supplementary to set up our equation and find the value for x.