Question

An architect designs two similar triangular patios. The first has angle measures of (x-7), (y+10), and 84. The second patio has angle measures of (x+30), 47, and 49. Find the value of x and y.

Answer 1

Since the sum of angles of a triangle is 180 degrees

Since the measures of the 3 angles of the 1st triangle are

$(x-7)^{\circ},(y+10)^{\circ},84^{\circ}$

Add them and equate the sum by 180 degrees

$x-7+y+10+84=180$

Add the like terms on the left side

$\begin{gathered} x+y+(-7+10+84)=180 \\ x+y+87=180 \end{gathered}$

Subtract 87 from both sides

$\begin{gathered} x+y+87-87=180-87 \\ x+y=93\rightarrow(1) \end{gathered}$

Since the measures of the 3 angles of the other triangles are

$(x+30)^{\circ},47^{\circ},49^{\circ}$

Add them and equate the sum by 180 degrees

$x+30+47+49=180$

Add the like terms on the left side

$\begin{gathered} x+(30+47+49)=180 \\ x+126=180 \end{gathered}$

Subtract 126 from both sides

$\begin{gathered} x+126-126=180-126 \\ x=54 \end{gathered}$

Substitute the value of x in equation (1) to find y

$54+y=93$

Subtract 54 from both sides

$\begin{gathered} 54-54+y=93-54 \\ y=39 \end{gathered}$

The answers are:

x = 54

y = 39