Question

In quadrilateral ABCD, the measure of angle A is half the sum of the measures of the other angles. What is the measure of angle A? Express your answer to the nearest integer.

Answer 1

Angle A measures 120°.

Explanation:

Step 1; Summing all the angles in a quadrilateral equals 360°. So the sum of angles A, B, C, and D will equal 360°. It is given that angle A is half the sum of angles B, C, D. Assume angle B =x, angle C = y, angle D = z.

Angle A = x+y+z / 2, take this as equation 1.

Step 2; Since ABCD is a quadrilateral,

x+y+z = 360, take this as equation 2.

Substituting equation 1 in 2, we get

(x+y+z)/2 + x+y+z = 360°,

Multiply 2 on the LHS due to LCM

(x+y+z) + 2 × (x+y+z) = 360°,

[(x+y+z)+ 2 × (x+y+z)]/2 = 360°.

So we get a denominator of 2 on the LHS, so we multiply the entire equation by 2,

3 × (x+y+z) = 360 × 2 = 720,

(Angle B + Angle C + Angle D) = 720 / 3 = 240.

So Angle B + Angle C + Angle D = 240,

Angle A is half 240°, So angle A equals 240°/2 = 120°.