Answer:
a.
m∠1 = 30°, because ∠1 and ∠ACB are vertical opposite angles
m∠A = 80°, calculation down
m∠D = 80°, calculation down
b.
The values of x and y are x = 7.5 and y = 8, calculation down
Step-by-step explanation:
a.
∵ AE intersects BD at point C
∴ m∠ACB = m∠1 ⇒ vertical opposite angles
∵ m∠ACB = 30°
∴ m∠1 = 30°
m∠1 = 30°, because ∠1 and angle ACB are vertical opposite angles
In Δ ABC
∵ m∠B = 70°
∵ m∠ACB = 30°
- The sum of the measures of the interior angles of a Δ is 180°
∴ m∠A + m∠B + m∠ACB = 180°
∵ m∠B = 70° and m∠ACB = 30°
∴ m∠A + 70 + 30 = 180
- Add the like terms
∴ m∠A + 100 = 180°
- Subtract 100 from both sides
∴ m∠A = 80°
In Δ DEC
∵ m∠E = 70°
∵ m∠DCE = 30°
- The sum of the measures of the interior angles of a Δ is 180°
∴ m∠D + m∠E + m∠DCE = 180°
∵ m∠E = 70° and m∠DCE = 30°
∴ m∠D + 70 + 30 = 180
- Add the like terms
∴ m∠D + 100 = 180°
- Subtract 100 from both sides
∴ m∠D = 80°
b.
In Δs ABC and DEC
∵ m∠A = m∠D
∵ m∠B = m∠E
∵ m∠ACB = m∠DCE
∴ The two triangles are similar by AAA postulate of similarity
- Their corresponding sides have equal ratio
∴
∵ AB = 6 and DE = y
∵ BC = x and EC = 10
∵ AC = 9 and DC = 12
- Substitute them in the ratio above
∴
- By using cross multiplication
∴ x × 12 = 10 × 9
∴ 12 x = 90
- Divide both sides by 12
∴ x = 7.5
∵ y × 9 = 6 × 12
∴ 9 y = 72
- Divide both sides by 9
∴ y = 8
The values of x and y are x = 7.5 and y = 8