Question

in triangle abc, the size of angle b is 5 times the size of angle a, and the size of angle c is 14° less than 4 times the size of angle a.
find the size of the angles.

the size of angle a is □°.
(simplify your answer. type an integer or a decimal.)
the size of angle b is □°.
(simplify your answer. type an integer or a decimal.)
the size of angle c is □°.
(simplify your answer. type an integer or a decimal.)

Explanation:

Step1: Define variables for angles

Let the measure of angle \( A \) be \( x \) degrees. Then angle \( B = 5x \) degrees (since it's 5 times angle \( A \)) and angle \( C = 4x - 14 \) degrees (14° less than 4 times angle \( A \)).

Step2: Use triangle angle sum property

In a triangle, the sum of interior angles is \( 180^\circ \). So, \( x + 5x + (4x - 14) = 180 \).

Step3: Simplify and solve for \( x \)

Combine like terms: \( x + 5x + 4x - 14 = 180 \) becomes \( 10x - 14 = 180 \).
Add 14 to both sides: \( 10x = 180 + 14 = 194 \).
Divide both sides by 10: \( x = \frac{194}{10} = 19.4 \).

Step4: Find measures of other angles

Angle \( B = 5x = 5 \times 19.4 = 97 \).
Angle \( C = 4x - 14 = 4 \times 19.4 - 14 = 77.6 - 14 = 63.6 \).

Answer:

The size of angle \( A \) is \( 19.4^\circ \).
The size of angle \( B \) is \( 97^\circ \).
The size of angle \( C \) is \( 63.6^\circ \).