Question

2) ABCD is a rectangle, answer each question and explain how you get the a

Answer 1

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 = $\frac{1}{2}$ AC

∴ DE = EB = $\frac{1}{2}$ DB

∵ DB = 12 units

∴ DE =  $\frac{1}{2}$  × 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