Question

Show all work provide a geometric equation

Answer 1

x = 11

y = 7

m∠ECH = 115°

Explanation:

An angle bisector is a line, ray, or segment that divides an angle into two congruent angles.

As ray CH bisects ∠FCG, then ∠GCH ≅ ∠HCF.

Equate the angles, and rearrange to isolate 2x:

`\begin{aligned}m \angle GCH &= m \angle HCF\\6x - 1 &= 4x + 3y\\6x - 1 - 4x &= 4x + 3y - 4x\\2x - 1 &= 3y\\2x - 1 + 1 &= 3y + 1\\2x &= 3y + 1\end{aligned}`

Angles on a straight line sum to 180°. Therefore, set the sum of ∠ECG, ∠GCH and ∠HCF to 180°:

`\begin{aligned}m \angle ECG + m \angle GCH + m \angle HCF &= 180^(\circ)\\\\7y + 1 + 6x - 1 + 4x + 3y &= 180\\\\10x +10y &= 180\\\\(10x+10y)/(10)&=(180)/(10)\\\\x + y &= 18\end{aligned}`

Rearrange the equation to isolate x:

`\begin{aligned}x+y&=18\\x+y-y&=18-y\\x &= 18 - y\end{aligned}`

Substitute the expression for x into the rearranged equation 2x = 3y + 1, and solve for y:

`\begin{aligned}2x&=3y+1\\2(18 - y) &= 3y + 1\\36 - 2y &= 3y + 1\\36-2y+2y&=3y+1+2y\\36&=5y+1\\36-1&=5y1-1\\35 &= 5y\\35/ 5&=5y / 5\\7&=y\\y &= 7\end{aligned}`

Therefore, the value of y is 7.

Substitute the found value of y into the expression for x:

`\begin{aligned}x&=18-y\\x &= 18 - 7\\x &= 11\end{aligned}`

Therefore, the value of x is 11.

As ∠ECH is the sum of ∠ECG and ∠GCH, to find an expression for ∠ECH, sum ∠ECG and ∠GCH:

`\begin{aligned}m \angle ECH &= m \angle ECG + m \angle GCH\\m \angle ECH &= 7y + 1 + 6x - 1\\m \angle ECH &= 7y + 6x\end{aligned}`

To calculate the measure of ∠ECH, we can substitute in the found values of x and y into the angle expression for ∠ECH:

`\begin{aligned}m \angle ECH &=7x+6y\\m \angle ECH &= 7(7) + 6(11)\\m \angle ECH &= 49 + 66\\m \angle ECH &= 115\end{aligned}`

Therefore, the measure of angle ECH is 115°.