Question

1. find the measure of each angle.

Explanation:

Step1: Use exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(14y−15=(y + 21)+(3y + 9)\).

Step2: Simplify the equation

$$
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Step3: Solve for \(y\)

Subtract \(4y\) from both sides: \(14y−4y−15=4y−4y + 30\), which gives \(10y−15 = 30\).
Add \(15\) to both sides: \(10y−15+15=30 + 15\), so \(10y=45\).
Divide both sides by \(10\): \(y=\frac{45}{10}=4.5\).

Step4: Find the measure of \(\angle A\)

Substitute \(y = 4.5\) into \(\angle A=(y + 21)^{\circ}\). Then \(\angle A=(4.5+21)^{\circ}=25.5^{\circ}\).

Step5: Find the measure of \(\angle B\)

Substitute \(y = 4.5\) into \(\angle B=(3y + 9)^{\circ}\). So \(\angle B=(3\times4.5+9)^{\circ}=(13.5 + 9)^{\circ}=22.5^{\circ}\).

Step6: Find the measure of \(\angle ACD\)

Substitute \(y = 4.5\) into \(\angle ACD=(14y−15)^{\circ}\). Then \(\angle ACD=(14\times4.5−15)^{\circ}=(63−15)^{\circ}=48^{\circ}\).

Answer:

\(\angle A = 25.5^{\circ}\), \(\angle B=22.5^{\circ}\), \(\angle ACD = 48^{\circ}\)