Answer
The values of angles numbered from 1 to 6 are \[\operatorname{m}\measuredangle 1=80{}^\circ \], \[\operatorname{m}\measuredangle 2=65{}^\circ \],\[\operatorname{m}\measuredangle 3=115{}^\circ \],\[\operatorname{m}\measuredangle 4=80{}^\circ \],\[\operatorname{m}\measuredangle 5=100{}^\circ \], and\[\operatorname{m}\measuredangle 6=80{}^\circ \].
Work Step by Step
In the given figure, there are two parallel lines l and m that are intersected by two traversals. In the given figure angle 2 and the angle given with a measure of \[115{}^\circ \]are supplementary angles and form a straight line. Hence, the measure of angle 2 is\[180{}^\circ -115{}^\circ =65{}^\circ \].
The angles formed by nonadjacent angles are called vertical angles. Thus, in the given figure angle formed by the traversals and angle given with a measure of \[35{}^\circ \]are vertical angles. The measure of vertical angles are equal.
The measure of angle formed in a triangle formed by two traversals is \[35{}^\circ \]. Other angle formed in the same triangle is the supplementary angle with an angle having a measure of\[115{}^\circ \]. The other angle is\[65{}^\circ \]. The sum of the measures of all angles in a triangle must be \[180{}^\circ \]
Compute the measure of angle 1 as shown below:
\[\begin{align}
& \operatorname{m}\measuredangle 1=180{}^\circ -\left( 65{}^\circ +35{}^\circ \right) \\
& =180{}^\circ -100{}^\circ \\
& =80{}^\circ
\end{align}\]
In the given figure, angle 6 and angle 1 are nonadjacent angles and thus called vertical angles. The measure of vertical angles are equal. Hence, the measure of angle 6 is\[80{}^\circ \].
The angles formed by the pair of angles that are on the exterior and opposite side of the traversal are called alternate exterior angles. Thus, in the given figure angle 3 and angle given with a measure of \[115{}^\circ \] are alternate exterior angle that are always of the same measure. Hence, the measure of angle 3 is \[115{}^\circ \].
The angles formed by the pair of angles that are on the inner and opposite side of the traversal are called alternate interior angles. Thus, in the given figure angle 6 and angle adjacent to angle 5 are alternate interior angle that are always of the same measure. The angle adjacent to angle 5 is \[80{}^\circ \].
In the given figure angle with a measure of \[80{}^\circ \]and the angle 5 are supplementary angles and form a straight line. Hence, the measure of angle 5 is \[180{}^\circ -80{}^\circ =100{}^\circ \].
In the given figure angle 4 and the angle 5 are supplementary angles and form a straight line. Hence, the measure of angle 4 is \[180{}^\circ -100{}^\circ =80{}^\circ \].
Hence, the values of angles numbered from 1 to 6 are \[\operatorname{m}\measuredangle 1=80{}^\circ \], \[\operatorname{m}\measuredangle 2=65{}^\circ \],\[\operatorname{m}\measuredangle 3=115{}^\circ \],\[\operatorname{m}\measuredangle 4=80{}^\circ \],\[\operatorname{m}\measuredangle 5=100{}^\circ \], and\[\operatorname{m}\measuredangle 6=80{}^\circ \].
