Find the largest and smallest angles of a convex quadrilateral if they are proportional

Find the largest and smallest angles of a convex quadrilateral if they are proportional to the numbers a) 2: 4: 5: 7; b) 3: 7: 4: 6.

Given:

ABCE – a quadrangle,

1) angle A: angle B: angle C: angle E = 2: 4: 5: 7;

2) angle A: angle B: angle C: angle E = 3: 7: 4: 6;

Find the degree measures of the angles of the quadrilateral ABCE: angle A, angle B, angle C, angle E -?

Solution:

Consider the quadrangle ABCE.

1) Let the degree measure of angle A be 2 * x degrees, then the degree measure of angle B is 4 * x degrees, the degree measure of angle C is 5 * x degrees, the degree measure of angle E is 7 * x degrees. We know that the sum of the degree measures of a parallelogram is 360 degrees. Let’s make the equation:

2 * x + 4 * x + 5 * x + 7 * x = 360;

x * (2 + 4 + 5 + 7) = 360;

x * 18 = 360;

x = 360: 18;

x = 20 degrees;

2 * 20 = 40 degrees – the degree measure of the angle A;

20 * 4 = 80 degrees – the degree measure of the angle B;

20 * 5 = 100 degrees – the degree measure of the angle C;

20 * 7 = 140 degrees – the degree measure of the angle E.

2) Let the degree measure of angle A be 3 * x degrees, then the degree measure of angle B is 7 * x degrees, the degree measure of angle C is 4 * x degrees, the degree measure of angle E is 6 * x degrees. We know that the sum of the degree measures of a parallelogram is 360 degrees. Let’s make the equation:

3 * x + 7 * x + 4 * x + 6 * x = 360;

x * (3 + 7 + 4 + 6) = 360;

x * 20 = 360;

x = 360: 20;

x = 18 degrees;

3 * 18 = 54 degrees – the degree measure of the angle A;

18 * 7 = 126 degrees – the degree measure of the angle B;

18 * 4 = 72 degrees – the degree measure of the angle C;

18 * 6 = 108 degrees – the degree measure of the angle E.

Answer: 1) 40 degrees; 80 degrees; 100 degrees; 140 degrees; 2) 54 degrees; 126 degrees; 108 degrees; 108 degrees.