Question

for questions 1 - 3, refer to the diagram at the right, where the m∠abc = 4x + 30, m∠abd = 5x + 10, and m∠dbc = 3x - 20. 1. find x. 2. find m∠abc. 3. find m∠abd. for questions 4 - 6, refer to the diagrams at the right, where ah bisects ∠mat.

Explanation:

Step1: Use angle - addition postulate

Since $\angle ABC=\angle ABD+\angle DBC$, we have the equation $4x + 30=(5x + 10)+(3x-20)$.

Step2: Simplify the right - hand side of the equation

$(5x + 10)+(3x-20)=5x+3x+10 - 20=8x - 10$. So the equation becomes $4x + 30=8x - 10$.

Step3: Solve for $x$

Subtract $4x$ from both sides: $30=8x-4x - 10$, which simplifies to $30 = 4x-10$. Then add 10 to both sides: $40 = 4x$. Divide both sides by 4, we get $x = 10$.

Step4: Find $m\angle ABC$

Substitute $x = 10$ into the expression for $m\angle ABC$. $m\angle ABC=4x + 30=4\times10+30=40 + 30=70$.

Step5: Find $m\angle ABD$

Substitute $x = 10$ into the expression for $m\angle ABD$. $m\angle ABD=5x + 10=5\times10+10=50 + 10=60$.

Answer:

1. $x = 10$
2. $m\angle ABC=70$
3. $m\angle ABD=60$