Question

1. find ∠1 if △xyz is regular. 15. find ∠4 if △xyz is regular. 16. find ∠4 if △xyz is not regular but ∠1 = 40° and m∠2 = 35°. 15. m∠1 + m∠2 + m∠3 = ___. 16. find the measure of each interior angle of a regular decagon. 17. find the measure of each exterior angle of a regular decagon.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. Also, for any polygon, the sum of exterior angles is $360^{\circ}$.

Step2: Solve question 14

A regular triangle (equilateral triangle) has all interior angles equal. Each interior angle of an equilateral triangle is $60^{\circ}$. $\angle1$ is an interior angle of an equilateral triangle $\triangle XYZ$, so $\angle1 = 60^{\circ}$.

Step3: Solve question 15

For a triangle, the sum of interior angles is $180^{\circ}$. So $m\angle1 + m\angle2 + m\angle3=180^{\circ}$.

Step4: Solve question 16

The sum of the interior angles of a triangle is $180^{\circ}$. Let the third - interior angle of $\triangle XYZ$ be $\angle3$. Then $\angle3=180^{\circ}-(40^{\circ} + 35^{\circ})=105^{\circ}$. $\angle4$ is an exterior angle to $\angle3$. By the exterior - angle property of a triangle (an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles), $\angle4=\angle1+\angle2 = 40^{\circ}+35^{\circ}=75^{\circ}$.

Step5: Solve question 16 (second part)

If $\triangle XYZ$ is regular, all interior angles are $60^{\circ}$. $\angle4$ is an exterior angle. An exterior angle of a triangle is supplementary to its adjacent interior angle. So $\angle4 = 120^{\circ}$.

Step6: Solve question 17

The sum of the exterior angles of any polygon is $360^{\circ}$. For a regular decagon ($n = 10$), each exterior angle $\theta=\frac{360^{\circ}}{n}$. Substituting $n = 10$, we get $\theta=\frac{360^{\circ}}{10}=36^{\circ}$.

Step7: Solve question 16 (first part)

We already found in Step 4 that when $\triangle XYZ$ is non - regular with $\angle1 = 40^{\circ}$ and $\angle2 = 35^{\circ}$, $\angle4=75^{\circ}$.

Step8: Solve question 16 (second - part for interior angles of regular triangle)

If $\triangle XYZ$ is regular, each interior angle is $60^{\circ}$.

Step9: Solve question 17 for interior angles of regular decagon

The formula for the measure of each interior angle of a regular polygon is $\alpha=\frac{(n - 2)\times180^{\circ}}{n}$. For a decagon ($n = 10$), $\alpha=\frac{(10 - 2)\times180^{\circ}}{10}=\frac{8\times180^{\circ}}{10}=144^{\circ}$.

Answer:

1. $\angle1 = 60^{\circ}$
2. $m\angle1 + m\angle2 + m\angle3 = 180^{\circ}$
3. Non - regular: $\angle4=75^{\circ}$; Regular: $\angle4 = 120^{\circ}$
4. Exterior angle of regular decagon: $36^{\circ}$; Interior angle of regular decagon: $144^{\circ}$