Answer:
m∠ABC = 90°, because the four angles are equal and their sum is 360°, then 360 ÷ 4 = 90°
DE = 6 units, because it is half DB which is equal to AC and AC = 12
m∠2 = 55°, because the sum of the measures of ∠1 and ∠2 is 90°
m∠4 = 75°, because it is equal m∠EAD and m∠2 in Δ AED which has ∠5
Step-by-step explanation:
Let us revise the properties of a triangle
- Each two opposite sides are equal and parallel
- Its two diagonals bisect each other and equal
- Its four angles are equal and the measure of each one is 90°
∵ ABCD is a rectangle
∴ m∠ABC = 90°
m∠ABC = 90°, because the four angles are equal and their sum is 360°, then 360 ÷ 4 = 90°
∵ AC and BD are the diagonals of the rectangle
∴ AC = BD
∵ AC = 12 units
∴ BD = 12 units
∵ The two diagonals bisect each other
∴ E is the mid-point of AC and BD
∴ AE = EC = [tex]\frac{1}{2}[/tex] AC
∴ DE = EB = [tex]\frac{1}{2}[/tex] DB
∵ DB = 12 units
∴ DE = [tex]\frac{1}{2}[/tex] × 12 = 6 units
DE = 6 units, because it is half DB which is equal to AC and AC = 12
∵ m∠D = 90° ⇒ property of rectangle
∵ m∠D = m∠1 + m∠2
∴ m∠1 + m∠2 = 90°
∵ m∠1 = 35°
∴ 35 + m∠2 = 90
- Subtract 35 from both sides
∴ m∠2 = 55°
m∠2 = 55°, because the sum of the measures of ∠1 and ∠2 is 90°
In Δ AED
∵ ED = EA ⇒ halves of equal diagonals are equal
∴ m∠EAD = m∠2 ⇒ base angles of an isosceles triangle
∵ m∠5 = 30°
- The sum of the interior angles of a triangle is 180°
∴ m∠2 + m∠2 + m∠5 = 180
∴ 2 m∠2 + 30 = 180
- Subtract 50 from both sides
∴ 2 m∠2 = 150
- Divide both sides by 2
∴ m∠2 = 75°
∴ m∠EAD = 75°
∵ AD // BC ⇒ opposite sides in a rectangle
∵ AC is a transversal
∴ m∠DAE = m∠4 ⇒ alternate angles
∴ m∠4 = 75°
m∠4 = 75°, because it is equal m∠EAD and m∠2 in Δ AED which has ∠5
