Question: Solve for $x$ and $y$, then find $m\angle ABC$
75°
$(2x - 5)°$
$y°$
A
B
C
Top-right graph: two intersecting lines, a horizontal line through B with arrows on both ends and a diagonal line through B from upper-left to lower-right with arrows on both ends; the angle at the upper-left is 75°, the upper-right angle is $(2x - 5)°$, and the lower-left angle is $y°$; point A is on the upper part of the diagonal, point C is on the right part of the horizontal; position_hint \in \{top-right, center\}
21. The perimeter of the rectangular figure is 85 cm, find the value of $x$. Show all work.
$x$
$13$ cm
$13$
1. **State the problem:** We are given three angles around point B formed by two intersecting lines: $75^\circ$, $(2x - 5)^\circ$, and $y^\circ$. We need to find $x$, $y$, and the measure of $\angle ABC$.
2. **Use the property of angles around a point:** The sum of angles around point B is $360^\circ$.
3. **Write the equation:**
$$75 + (2x - 5) + y + 180 = 360$$
Here, $180^\circ$ is the straight angle formed by the horizontal line through B.
4. **Simplify the equation:**
$$75 + 2x - 5 + y + 180 = 360$$
$$(75 - 5 + 180) + 2x + y = 360$$
$$250 + 2x + y = 360$$
5. **Isolate $2x + y$:**
$$2x + y = 360 - 250$$
$$2x + y = 110$$
6. **Use vertical angles property:** The angle $y$ is vertical to the $75^\circ$ angle, so
$$y = 75$$
7. **Substitute $y = 75$ into the equation:**
$$2x + 75 = 110$$
8. **Solve for $x$:**
$$2x = 110 - 75$$
$$2x = 35$$
$$x = \frac{35}{2} = 17.5$$
9. **Find $m\angle ABC$:** The angle $m\angle ABC$ is the angle $(2x - 5)^\circ$.
10. **Calculate $m\angle ABC$:**
$$m\angle ABC = 2(17.5) - 5 = 35 - 5 = 30^\circ$$
**Final answers:**
- $x = 17.5$
- $y = 75$
- $m\angle ABC = 30^\circ$