Question

1. consider the triangle shown. for each angle measure given, determine whether you can use this triangle to draw each of the following angles, without using a protractor. if yes, describe how you would create the angle. if no, describe why you cannot create the angle.

a 72°
b 88°
c 110°
d 144°

Explanation:

First, calculate the third angle of the given triangle to know all its angles:

Step1: Find triangle's third angle

Sum of angles in triangle is $180^\circ$.
$$180^\circ - 52^\circ - 36^\circ = 92^\circ$$
The triangle has angles $36^\circ$, $52^\circ$, $92^\circ$.

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Step2: Analyze $72^\circ$ angle

$72^\circ = 36^\circ + 36^\circ$.
We can copy the $36^\circ$ angle twice, adjacent to each other, to form $72^\circ$.

Step3: Analyze $88^\circ$ angle

$88^\circ = 36^\circ + 52^\circ$.
We can copy the $36^\circ$ and $52^\circ$ angles adjacent to each other to form $88^\circ$.

Step4: Analyze $110^\circ$ angle

$110^\circ$ cannot be expressed as a sum or difference of $36^\circ$, $52^\circ$, $92^\circ$ (or combinations that form a constructible angle with the triangle). No combination of the triangle's angles equals or can form $110^\circ$.

Step5: Analyze $144^\circ$ angle

$144^\circ = 180^\circ - 36^\circ$.
We can draw a straight line ($180^\circ$) and copy the $36^\circ$ angle inside it; the remaining angle is $144^\circ$.

Answer:

a. Yes: Copy the $36^\circ$ angle twice, adjacent to one another, to form $72^\circ$.
b. Yes: Copy the $36^\circ$ and $52^\circ$ angles adjacent to one another to form $88^\circ$.
c. No: $110^\circ$ cannot be formed by adding, subtracting, or combining the triangle's angles ($36^\circ$, $52^\circ$, $92^\circ$).
d. Yes: Draw a straight $180^\circ$ angle, then copy the $36^\circ$ angle inside it; the remaining angle is $144^\circ$.