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Describing Angles

Describing Angles 1.14 - Solution

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The angle yy and the angle which is 3131 ^\circ are vertical angles. According to the Vertical Angles Theorem the angles are congruent and have the same measure.

Exercise 663 2.svg

The sum of the green angles is 31+31=62.31 ^\circ+31^\circ=62^\circ. Since a complete rotation is 360,360 ^\circ, the remaining 298298 ^\circ are distributed over the other two unknown angles. These also form a pair of vertical angles and are congruent. Thus, their measure is half of 298.298^\circ. 2982=149 \dfrac{298^\circ}{2\,\,}=149^\circ Thus, both x\angle x and z\angle z have the measure 149.149^\circ.

Exercise 663 3.svg

Alternative Solution

Alternative Solution

As in the other solution, we can determine the mym\angle y to be 31.31^\circ.

Exercise 663 4.svg

The x\angle x and any of the angles with the measure 3131^\circ are supplementary angles. Thus, the sum of their measures is 180.180^\circ. x+31=180x=149. x+31^\circ=180^\circ \quad \Leftrightarrow \quad x=149^\circ. xx and zz are vertical angles and are congruent. Therefore, mz=149.m\angle z=149^\circ.