Question

1. if $mangle ecb = 125^{circ}$, $mangle ecd = 2x - 2$, and $mangle dcb = 4x + 1$, find each value.
2. if $mangle ecd = 6x - 1$ and $mangle dcb = 7x - 14$ find each value.
3. if $mangle ecd = 7x + 20$ and $mangle acb = 2x+80$ find each value.

Explanation:

Step1: Use angle - addition property

Since $\angle ECB=\angle ECD+\angle DCB$, we have the equation $125=(2x - 2)+(4x + 1)$.

Step2: Simplify the right - hand side of the equation

Combine like terms: $(2x - 2)+(4x + 1)=2x+4x-2 + 1=6x-1$. So, $6x-1 = 125$.

Step3: Solve for $x$

Add 1 to both sides of the equation: $6x-1+1=125 + 1$, which gives $6x=126$. Then divide both sides by 6: $x=\frac{126}{6}=21$.

Step4: Find $\angle ECD$

Substitute $x = 21$ into the expression for $\angle ECD$: $m\angle ECD=2x-2=2\times21-2=42 - 2=40^{\circ}$.

Step5: Find $\angle DCB$

Substitute $x = 21$ into the expression for $\angle DCB$: $m\angle DCB=4x + 1=4\times21+1=84 + 1=85^{\circ}$.

Step1: Use angle - addition property

Since $\angle ECD+\angle DCB=\angle ECB$ and $\angle ECD = 8x-1$, $\angle DCB=7x - 14$, we have the equation $(8x-1)+(7x - 14)=\angle ECB$. Assuming $\angle ECB$ is a straight - angle ($180^{\circ}$) (as no other information about $\angle ECB$ is given and the angles seem to be on a straight line), $8x-1+7x-14 = 180$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $8x+7x-1-14=15x-15$. So, $15x-15 = 180$.

Step3: Solve for $x$

Add 15 to both sides: $15x-15 + 15=180+15$, which gives $15x=195$. Then divide both sides by 15: $x=\frac{195}{15}=13$.

Step4: Find $\angle ECD$

Substitute $x = 13$ into the expression for $\angle ECD$: $m\angle ECD=8x-1=8\times13-1=104 - 1=103^{\circ}$.

Step5: Find $\angle DCB$

Substitute $x = 13$ into the expression for $\angle DCB$: $m\angle DCB=7x - 14=7\times13-14=91 - 14=77^{\circ}$.

Step1: Use angle - addition property

Since $\angle ECD+\angle DCB=\angle ECB$ and $\angle ECD = 7x+20$, $\angle DCB=2x + 80$, and assuming $\angle ECB$ is a straight - angle ($180^{\circ}$), we have the equation $(7x+20)+(2x + 80)=180$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $7x+2x+20 + 80=9x+100$. So, $9x+100 = 180$.

Step3: Solve for $x$

Subtract 100 from both sides: $9x+100-100=180 - 100$, which gives $9x=80$. Then $x=\frac{80}{9}\approx8.89$.

Step4: Find $\angle ECD$

Substitute $x=\frac{80}{9}$ into the expression for $\angle ECD$: $m\angle ECD=7x+20=7\times\frac{80}{9}+20=\frac{560}{9}+20=\frac{560 + 180}{9}=\frac{740}{9}\approx82.22^{\circ}$.

Step5: Find $\angle DCB$

Substitute $x=\frac{80}{9}$ into the expression for $\angle DCB$: $m\angle DCB=2x + 80=2\times\frac{80}{9}+80=\frac{160}{9}+80=\frac{160+720}{9}=\frac{880}{9}\approx97.78^{\circ}$.

Answer:

$x = 21$
$m\angle ECD=40^{\circ}$
$m\angle DCB=85^{\circ}$