Question

1. start with the triangle sum theorem.$m\angle a + m\angle b + m\angle c = 180^\circ$2. plug in the values you know.$\\_\\_\\_\\_^\circ + (\\_\\_\\_\\_)^\circ + \\_\\_\\_\\_^\circ = 180^\circ$3. simplify and solve for $x$.$5x + 30 = 180LXB0x = \\_\\_\\_\\_$4. use that value of $x$ to find $m\angle a$.$m\angle a = (2x)^\circLXB1= \\_\\_\\_\\_^\circ$5. repeat step 4 for $m\angle b$.$m\angle b = (3x - 10)^\circLXB2= 80^\circ$6. use the triangle sum theorem to check your answers.$m\angle a + m\angle b + m\angle c = 180^\circLXB3180^\circ = 180^\circ$

Explanation:

Step1: Identify known angle values

From steps 3, 4, 5, we infer: $m\angle A=(2x)^\circ$, $m\angle B=(3x-10)^\circ$, $m\angle C=40^\circ$. Substitute into the theorem:
$$2x + (3x-10) + 40 = 180$$

Step2: Solve for $x$

Simplify and isolate $x$:
$$5x + 30 = 180$$
$$5x = 150$$
$$x = 30$$

Step3: Calculate $m\angle A$

Substitute $x=30$ into $m\angle A=(2x)^\circ$:
$$m\angle A=2(30)^\circ=60^\circ$$

Step4: Verify $m\angle B$

Substitute $x=30$ into $m\angle B=(3x-10)^\circ$:
$$m\angle B=(3\cdot30 - 10)^\circ=80^\circ$$

Step5: Check with triangle sum theorem

Substitute all angle values:
$$60 + 80 + 40 = 180$$
$$180=180$$

Answer:

Step 2 blanks: $\boldsymbol{2x}$, $\boldsymbol{3x-10}$, $\boldsymbol{40}$
Step 3 blank: $\boldsymbol{30}$
Step 4 blanks: $\boldsymbol{30}$, $\boldsymbol{60}$
Step 5 blank: $\boldsymbol{30}$
Step 6 blanks: $\boldsymbol{60}$, $\boldsymbol{80}$
Final values: $x=30$, $m\angle A=60^\circ$, $m\angle B=80^\circ$