Question

1-99. calculate all the missing angle measures in the diagrams below. homework help
a.
b.
1-101. write the area of the rectangle below as the area as a product equal to the area as a sum. homework help

Explanation:

Step 1: Solve for missing angles in part a

Use the fact that the sum of angles around a point is 360° and vertical - angles are equal. Let the missing angles be \(x\) and \(y\). The angle opposite the 75° angle is 75° (vertical angles), and the angle opposite the 85° angle is 85° (vertical angles). So \(x = 360-(75 + 75+85 + 85)=40\)°. The other non - labeled angles are 75°, 85°, 75°, and 85° due to vertical angles.

Step 2: Solve for missing angles in part b

First, use the linear - pair property. The angle adjacent to 110° in the triangle is \(180 - 110=70\)°. In the triangle, using the angle - sum property of a triangle (\(180^{\circ}\) for the sum of interior angles of a triangle), if one angle is 70° and another is 30°, then the third angle in the triangle is \(180-(70 + 30)=80\)°. The vertical angles to these non - 110° angles are also 70°, 30°, and 80°. The other angles formed by the intersecting lines can be found using linear - pair and vertical - angle relationships.

Step 3: Solve for the area of the rectangle in 1 - 101

Count the number of \(x^2\), \(x\), and unit squares. There are 2 \(x^2\) terms, 8 \(x\) terms. The length of the rectangle can be written as \(2x + 4\) and the width as \(x+2\). The area as a product is \((2x + 4)(x + 2)\). Expanding \((2x+4)(x + 2)=2x\times x+2x\times2+4\times x + 4\times2=2x^{2}+4x+4x + 8=2x^{2}+8x + 8\). The area as a sum is \(2x^{2}+8x+8\).

Answer:

a. The missing angles are found as described above with non - labeled angles being 75°, 85°, 75°, and 85° and the remaining angle being 40°.
b. The missing angles are 70°, 30°, 80° and their vertical - angle counterparts.
1 - 101: Area as a product: \((2x + 4)(x + 2)\), Area as a sum: \(2x^{2}+8x + 8\)