Question: 17. . If line values of $x$ and $y$ in the diagram below.
$(7x - 31)^\circ$
$63^\circ$
$(5x - 8)^\circ$
$(4y + 27)^\circ$
+ 36
Graph/shape: A triangle is drawn with a top horizontal segment and two slanted sides meeting at a bottom vertex; two parallel slanted rays labeled $l$ and $m$ extend upward-left and upward-right from the top corners. position_hint=center
how to solve
1. **State the problem:** We are given a triangle with angles expressed in terms of $x$ and $y$ and some angle measures. We need to find the values of $x$ and $y$.
2. **Identify the angles and relationships:** The triangle has three angles, and the sum of the interior angles of any triangle is always $$180^\circ$$.
3. **Write the equation for the triangle's angles:**
The angles given are:
- $(7x - 31)^\circ$
- $63^\circ$
- $(5x - 8)^\circ$
So, the sum is:
$$ (7x - 31) + 63 + (5x - 8) = 180 $$
4. **Simplify the equation:**
$$ 7x - 31 + 63 + 5x - 8 = 180 $$
$$ (7x + 5x) + (-31 + 63 - 8) = 180 $$
$$ 12x + 24 = 180 $$
5. **Solve for $x$:**
$$ 12x + 24 = 180 $$
Subtract 24 from both sides:
$$ 12x + \cancel{24} - \cancel{24} = 180 - 24 $$
$$ 12x = 156 $$
Divide both sides by 12:
$$ \frac{12x}{\cancel{12}} = \frac{156}{\cancel{12}} $$
$$ x = 13 $$
6. **Use $x$ to find $y$:**
We have another angle expression involving $y$:
$$(4y + 27)^\circ + 36^\circ$$
Since the problem mentions parallel rays $l$ and $m$, these angles likely form a linear pair or supplementary angles summing to $$180^\circ$$.
Assuming:
$$ (4y + 27) + 36 = 180 $$
7. **Solve for $y$:**
$$ 4y + 27 + 36 = 180 $$
$$ 4y + 63 = 180 $$
Subtract 63 from both sides:
$$ 4y + \cancel{63} - \cancel{63} = 180 - 63 $$
$$ 4y = 117 $$
Divide both sides by 4:
$$ \frac{4y}{\cancel{4}} = \frac{117}{\cancel{4}} $$
$$ y = 29.25 $$
**Final answers:**
$$ x = 13 $$
$$ y = 29.25 $$