Question

find m∠abd and m∠cbd. 25. m∠abc = 77°. 26. m∠abc = 111°. (3x + 22)°. (5x - 17)°. (-10x + 58)°. (6x + 41)°

Explanation:

Step1: Use angle - addition postulate

Since $\angle ABC=\angle ABD+\angle CBD$, for problem 25, we have $(3x + 22)+(5x-17)=77$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $3x+5x+22 - 17=8x + 5$. So, $8x+5 = 77$.

Step3: Solve for $x$

Subtract 5 from both sides: $8x=77 - 5=72$. Then divide both sides by 8, $x = 9$.

Step4: Find $\angle ABD$ and $\angle CBD$

For $\angle ABD$, substitute $x = 9$ into $3x + 22$, we get $3\times9+22=27 + 22=49^{\circ}$.
For $\angle CBD$, substitute $x = 9$ into $5x-17$, we get $5\times9-17=45 - 17 = 28^{\circ}$.

For problem 26:

Step1: Use angle - addition postulate

Since $\angle ABC=\angle ABD+\angle CBD$, we have $(-10x + 58)+(6x+41)=111$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $-10x+6x+58 + 41=-4x+99$. So, $-4x + 99=111$.

Step3: Solve for $x$

Subtract 99 from both sides: $-4x=111 - 99 = 12$. Then divide both sides by - 4, $x=-3$.

Step4: Find $\angle ABD$ and $\angle CBD$

For $\angle ABD$, substitute $x=-3$ into $-10x + 58$, we get $-10\times(-3)+58=30 + 58 = 88^{\circ}$.
For $\angle CBD$, substitute $x=-3$ into $6x + 41$, we get $6\times(-3)+41=-18 + 41 = 23^{\circ}$.

Answer:

1. $m\angle ABD = 49^{\circ}$, $m\angle CBD=28^{\circ}$
2. $m\angle ABD = 88^{\circ}$, $m\angle CBD = 23^{\circ}$