Question

4. Use the given information of the triangle answer the following questions.

Answer 1

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

∴ $\frac{AB}{DE}=\frac{BC}{EC}=\frac{AC}{DC}$

∵ AB = 6 and DE = y

∵ BC = x and EC = 10

∵ AC = 9 and DC = 12

- Substitute them in the ratio above

∴ $\frac{6}{y}=\frac{x}{10}=\frac{9}{12}$

- 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