Question

1. m∠abc = 61°. find m∠abd and m∠dbc. 3. ∠abc = 115°. find m∠abd and m∠cbd

Explanation:

Question 2

Step1: Set up equation

Since $m\angle ABC=m\angle ABD + m\angle DBC$, we have $(3x + 22)+(5x-17)=61$.

Step2: Simplify left - hand side

Combine like terms: $3x+5x+22 - 17=8x + 5$. So, $8x+5 = 61$.

Step3: Solve for x

Subtract 5 from both sides: $8x=61 - 5=56$. Then divide by 8, $x = 7$.

Step4: Find $m\angle ABD$

Substitute $x = 7$ into $m\angle ABD=3x + 22$. So, $m\angle ABD=3\times7+22=21 + 22=37^{\circ}$.

Step5: Find $m\angle DBC$

Substitute $x = 7$ into $m\angle DBC=5x-17$. So, $m\angle DBC=5\times7-17=35 - 17=24^{\circ}$.

Question 3

Step1: Set up equation

Since $m\angle ABC=m\angle ABD + m\angle DBC$, we have $(-10x + 58)+(6x + 41)=115$.

Step2: Simplify left - hand side

Combine like terms: $-10x+6x+58 + 41=-4x+99$. So, $-4x + 99=115$.

Step3: Solve for x

Subtract 99 from both sides: $-4x=115 - 99 = 16$. Then divide by - 4, $x=-4$.

Step4: Find $m\angle ABD$

Substitute $x=-4$ into $m\angle ABD=-10x + 58$. So, $m\angle ABD=-10\times(-4)+58=40 + 58=28^{\circ}$.

Step5: Find $m\angle DBC$

Substitute $x=-4$ into $m\angle DBC=6x + 41$. So, $m\angle DBC=6\times(-4)+41=-24 + 41=87^{\circ}$.

Answer:

$m\angle ABD = 37^{\circ}$, $m\angle DBC=24^{\circ}$