Question

18. Find m/BCA.
(9x + 1)
A
(5x + 12)
B
(10x-37)
C

Answer 1

∠BCA 57 degrees

Explanation:

∠ABC + (5x + 12) = 180

=> ∠ABC = 180 - (5x + 12)

∠BAC + (9x + 1) = 180

=> ∠BAC = 180 - (9x + 1)

∠BCA + (10x - 37) = 180

=> ∠BCA = 180 - (10x - 37)

∠ABC + ∠BAC + ∠BCA = 180

Substitute to find x

[180 - (5x + 12)] + [180 - (9x + 1)] + [180 - (10x - 37)] = 180

180 + 180 + 180 - 5x - 9x - 10x - 12 - 1 + 37 = 180

564 - 24x = 180

24x = 564 - 180

24x = 384

x = 384/24 = 16

Substitue x = 16

∠BCA = 180 - (10x - 37)

∠BCA = 180 - 10(16) + 37 = 57

Answer 2

m∠BCA = 57°

Explanation:

To find the measure of angle BCA we must first find the value of x.

The diagram gives expressions for the exterior angles of the triangle.

The exterior angles of a triangle sum to 360°. Therefore, to calculate the value of x, equate the sum of the exterior angles to 360° and solve for x.

⇒ (9x + 1)° + (5x + 12)° + (10x - 37)° = 360°

⇒ 9x + 1 + 5x + 12 + 10x - 37 = 360

⇒ 9x + 5x + 10x + 1 + 12 - 37 = 360

⇒ 24x - 24 = 360

⇒ 24x - 24 + 24 = 360 + 24

⇒ 24x = 384

⇒ 24x ÷ 24 = 384 ÷ 24

⇒ x = 16

Each interior and exterior angle of a triangle form a linear pair.

As the sum of angles of a linear pair is always equal to 180°, to find the measure of angle BCA, equate the sum of ∠BCA and its exterior angle to 180°:

⇒ (10x - 37)° + m∠BCA = 180°

Substitute the found value of x and solve for the angle:

⇒ (10(16) - 37)° + m∠BCA = 180°

⇒ (160 - 37)° + m∠BCA = 180°

⇒ 123° + m∠BCA = 180°

⇒ 123° + m∠BCA - 123° = 180° - 123°

⇒ m∠BCA = 57°

Therefore, the measure of angle BCA is 57°.