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. 18. deductive or inductive? kawika notices that spaghetti has been on the school menu for the past five wednesdays. kawika decides that the school always serves spaghetti on wednesdays. 19. deductive or inductive? by using the definitions of equilateral triangle and of perimeter, elena concludes that the perimeter of every equilateral is three times the length of a side. 22. an equilateral triangle is _ an equiangular triangle. complete each statement with the word always, sometimes, or never. 23. a scalene triangle _ has all acute angles. 24. a scalene triangle _ has three different lengths. 25. a scalene triangle is ___ isosceles.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$, i.e., for $\triangle XYZ$, $m\angle1 + m\angle2+m\angle3 = 180^{\circ}$.

Step2: Recall formula for interior and exterior angles of a regular polygon

The formula for the measure of an interior angle $\theta_i$ of a regular $n$-sided polygon is $\theta_i=\frac{(n - 2)\times180^{\circ}}{n}$, and the measure of an exterior angle $\theta_e$ of a regular $n$-sided polygon is $\theta_e=\frac{360^{\circ}}{n}$. For a decagon ($n = 10$), the measure of an interior angle is $\theta_i=\frac{(10 - 2)\times180^{\circ}}{10}=144^{\circ}$, and the measure of an exterior angle is $\theta_e=\frac{360^{\circ}}{10}=36^{\circ}$.

Step3: Recall triangle - type properties

A scalene triangle has three different side - lengths. An equilateral triangle is always equiangular (all angles are equal, each $60^{\circ}$). A scalene triangle can have all acute angles.

Step4: Solve for angles in non - regular triangle

If $\triangle XYZ$ is not regular, $\angle1 = 40^{\circ}$ and $\angle2=35^{\circ}$, and using the angle - sum property of a triangle $m\angle1 + m\angle2+m\angle3 = 180^{\circ}$, then $m\angle3=180^{\circ}-(40^{\circ}+35^{\circ}) = 105^{\circ}$.

Step5: Determine deductive/inductive reasoning

Inductive reasoning is based on patterns and observations. Kawika notices spaghetti on the school menu for the past five Wednesdays and concludes it will be on the menu next Wednesday, which is inductive reasoning. Deductive reasoning uses definitions and facts. Elena concludes that the perimeter of an equilateral triangle is three times the length of a side using the definition of an equilateral triangle, which is deductive reasoning.

Answer:

1. If $\triangle XYZ$ is regular, each interior angle is $60^{\circ}$, so if $\angle1$ is an interior angle of a regular $\triangle XYZ$, $m\angle1 = 60^{\circ}$.
2. If $\triangle XYZ$ is regular, each interior angle is $60^{\circ}$. If $\triangle XYZ$ is not regular, $\angle1 = 40^{\circ}$, $\angle2 = 35^{\circ}$, then $m\angle3=105^{\circ}$ and we cannot directly find $\angle4$ without more information about its relation to the triangle. Assuming $\angle4$ is an exterior angle adjacent to $\angle3$, then $m\angle4=180^{\circ}-105^{\circ}=75^{\circ}$.
3. Measure of each interior angle of a regular decagon: $\frac{(10 - 2)\times180^{\circ}}{10}=144^{\circ}$. Measure of each exterior angle of a regular decagon: $\frac{360^{\circ}}{10}=36^{\circ}$.
4. Inductive: Kawika's reasoning. Deductive: Elena's reasoning.
5. Never. An equilateral triangle is always equiangular.
6. A scalene triangle has three different side - lengths. It can have all acute angles.
7. A scalene triangle has three different side - lengths.
8. A scalene triangle is not isosceles.