Question

1. answer the following for the given figure.

(lesson 4.2) (1 point)
a. suppose \\( \angle 1 \\) is \\( 105 ^ { \circ } \\). list all the angles
congruent to \\( \angle 1 \\).
b. suppose \\( \angle 3 \\) is also \\( 105 ^ { \circ } \\). name all angles
supplementary to \\( \angle 3 \\). what is their
measurement?

Explanation:

Step1: Identify congruent angles for part a

Congruent angles are angles that have the same measure. Vertical angles are congruent. Also, if two parallel lines are cut by a transversal, corresponding angles are congruent.

• $\angle1$ and $\angle3$ are corresponding angles (if the lines are parallel, which is implied by the figure's structure).
• $\angle1$ and $\angle5$: $\angle1$ and $\angle3$ are congruent (corresponding), $\angle3$ and $\angle5$ are vertical angles (so congruent).
• $\angle1$ and $\angle7$: $\angle1$ and $\angle3$ (corresponding), $\angle3$ and $\angle5$ (vertical), $\angle5$ and $\angle7$ (corresponding).

Step2: Identify supplementary angles for part b

Supplementary angles add up to $180^{\circ}$.

• $\angle3$ and $\angle4$: They form a linear - pair (so supplementary).
• $\angle3$ and $\angle6$: Vertical angles of $\angle4$ and $\angle3$ (since $\angle4$ and $\angle6$ are vertical, and $\angle3+\angle4 = 180^{\circ}$, then $\angle3+\angle6=180^{\circ}$).
• $\angle3$ and $\angle2$: $\angle2$ and $\angle4$ are vertical (so $\angle2=\angle4$), and $\angle3+\angle4 = 180^{\circ}$.
• $\angle3$ and $\angle8$: $\angle8$ and $\angle2$ are vertical (so $\angle8 = \angle2$), and $\angle3+\angle2=180^{\circ}$.

The measure of the supplementary angles: Let the measure of the supplementary angle be $x$. Using the formula for supplementary angles $x + 105^{\circ}=180^{\circ}$, so $x=180^{\circ}-105^{\circ}=75^{\circ}$.

Answer:

a. $\angle3,\angle5,\angle7$
b. Angles: $\angle2,\angle4,\angle6,\angle8$; Measure: $75^{\circ}$