A parallelogram has area 85cm squared. The side lengths are 10 cm and 9 cm. What are the measures of the interior angles?
draw it with 110 as base
base*height = area
h = 85/10 = 8.5
sin A = 8.5/9
compute A
To find the measures of the interior angles of a parallelogram, we can use the fact that opposite angles in a parallelogram are equal.
Step 1: Calculate the height of the parallelogram using the formula for the area of a parallelogram: area = base × height.
85 cm² = 10 cm × height
Height = 85 cm² / 10 cm
Height = 8.5 cm
Step 2: Calculate the measure of one of the interior angles using the formula for the area of a parallelogram: area = base × height, where the base is one of the side lengths.
85 cm² = 9 cm × 8.5 cm
85 cm² = 76.5 cm²
Area = 85 cm² (given)
Therefore, one of the interior angles measures 85 degrees.
Step 3: Since opposite angles in a parallelogram are equal, the other interior angle also measures 85 degrees.
Therefore, the measures of the interior angles of the parallelogram are 85 degrees each.
To find the measures of the interior angles of a parallelogram, we need to know the length of the sides and the area.
Let's start by finding the height (h) of the parallelogram using the formula for its area:
Area = base × height
Since we know that the area is 85 cm² and the base is 9 cm, we can rearrange the formula to solve for the height:
85 cm² = 9 cm × h
Dividing both sides by 9 cm:
h = 85 cm² / 9 cm
h ≈ 9.44 cm
Now that we know the height, we can use trigonometry to find the measures of the interior angles. In a parallelogram, opposite angles are congruent, so we only need to find one angle.
Let's consider one of the smaller triangles formed by the height and the side lengths of the parallelogram:
Using the Pythagorean theorem, we can find the third side (b) of the triangle:
b² = (10 cm)² - (9.44 cm)²
b ≈ √(100 cm² - 89.4336 cm²)
b ≈ √10.5664 cm²
b ≈ 3.25 cm
Now, we can find the measure of one of the interior angles of the parallelogram by using the inverse trigonometric function:
∠A = arctan(h/b)
∠A = arctan(9.44 cm / 3.25 cm)
∠A ≈ arctan(2.9)
Using a calculator or a trigonometric table, we can find that arctan(2.9) ≈ 70.53°.
Since opposite angles in a parallelogram are congruent, each interior angle is 70.53°.