Question

an architect designs two similar triangular patios. the first patio has angle measures of $(x - 5)^{circ}$, $(y + 20)^{circ}$, and $50^{circ}$. the second patio has angle measures of $(x - 15)^{circ}$, $60^{circ}$, and $70^{circ}$. find the values of $x$ and $y$.
$x = square$
$y = square$

Explanation:

Step1: Find the value of \(x\)

Since the two triangles are similar, their corresponding angles are equal.
We know that \((x - 15)^{\circ}\) in the second - patio must be equal to \((x - 5)^{\circ}\) or \((y + 20)^{\circ}\) or \(50^{\circ}\).
The sum of angles in a triangle is \(180^{\circ}\). For the second patio, \(60^{\circ}+70^{\circ}+(x - 15)^{\circ}=180^{\circ}\)

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Step2: Find the value of \(y\)

Substitute \(x = 65\) into the first - patio's angle \((x - 5)^{\circ}=(65 - 5)^{\circ}=60^{\circ}\)
Since the sum of angles in a triangle is \(180^{\circ}\) for the first patio: \((x - 5)^{\circ}+(y + 20)^{\circ}+50^{\circ}=180^{\circ}\)
Substitute \(x = 65\) (we found \(x - 5=60\)) into the equation: \(60+(y + 20)+50 = 180\)

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Answer:

\(x = 65\), \(y = 50\)