Question: CHAPTER 2 1 Geometry Lines and Angles DIRECTIONS: Find the value of each expression in degrees. For A through E, use the following figure: A. what is the measure of $\angle a$ ? B. what is the measure of $b$ ? C. what is the measure of $c + d$ ? D. what is the measure of $e$ ? E. What is the measure of $2c + 3f$ ? position_hint: top-right A line labeled $p$ slopes down from left to right and intersects the top horizontal line at an angle marked $40^\circ$; nearby are angles $a$ and $b$ on the first and second horizontal lines, with $c$ on the right-side diagonal intersection on the second horizontal line, and a small right-angle marker at the lower central intersection where lines $m$, $n$, and the diagonal lines meet; another diagonal labeled $q$ intersects the top line with angles $f$ and $g$, and the bottom cluster has angles $d$ and $e$ around the intersection.

1. **State the problem:** We need to find the measures of angles $a$, $b$, $c + d$, $e$, and $2c + 3f$ in degrees based on the given figure with lines and angles. 2. **Given information:** - Angle at the intersection of line $p$ with the top horizontal line is $40^\circ$. - There is a right angle ($90^\circ$) at the lower central intersection. 3. **Find $a$:** - Since $p$ intersects the top horizontal line at $40^\circ$, and $a$ is the adjacent angle on the straight line, angles $a$ and $40^\circ$ are supplementary. - Use the supplementary angle rule: $$a + 40 = 180$$ - Solve for $a$: $$a = 180 - 40 = 140$$ 4. **Find $b$:** - $a$ and $b$ are corresponding angles formed by parallel lines and a transversal, so $b = a = 140$. 5. **Find $c + d$:** - The right angle marker indicates $c + d = 90^\circ$. 6. **Find $e$:** - $e$ and $d$ are complementary to the right angle, so $e = 90 - d$. - Since $c + d = 90$, then $d = 90 - c$. - Substitute $d$ into $e = 90 - d$: $$e = 90 - (90 - c) = c$$ 7. **Find $2c + 3f$:** - $f$ is vertically opposite to the $40^\circ$ angle, so $f = 40$. - Calculate: $$2c + 3f = 2c + 3 \times 40 = 2c + 120$$ **Final answers:** - $a = 140^\circ$ - $b = 140^\circ$ - $c + d = 90^\circ$ - $e = c$ - $2c + 3f = 2c + 120$