Question

remember
for problems 12 and 13, solve for x.

1. $0.7(x+10)-4.7=6.5$
2. $-0.5(x-8)+3.8=-11.5$
3. determine whether $\triangle tuv$ is similar to $\triangle kml$ by the angle-angle criterion. explain how you know.

Explanation:

Problem 12: Solve for $x$

Step1: Isolate the term with $x$

Add 4.7 to both sides.
$0.7(x+10) = 6.5 + 4.7$
$0.7(x+10) = 11.2$

Step2: Divide to simplify

Divide both sides by 0.7.
$x+10 = \frac{11.2}{0.7}$
$x+10 = 16$

Step3: Solve for $x$

Subtract 10 from both sides.
$x = 16 - 10$

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Problem 13: Solve for $x$

Step1: Isolate the term with $x$

Subtract 3.8 from both sides.
$-0.5(x-8) = -11.5 - 3.8$
$-0.5(x-8) = -15.3$

Step2: Divide to simplify

Divide both sides by -0.5.
$x-8 = \frac{-15.3}{-0.5}$
$x-8 = 30.6$

Step3: Solve for $x$

Add 8 to both sides.
$x = 30.6 + 8$

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Problem 14: Check triangle similarity

Step1: Find $\angle K$ in $\triangle KML$

Sum of angles in a triangle is $180^\circ$.
$\angle K = 180^\circ - 112^\circ - 40^\circ = 28^\circ$

Step2: Find $\angle V$ in $\triangle TUV$

Sum of angles in a triangle is $180^\circ$.
$\angle V = 180^\circ - 112^\circ - 30^\circ = 38^\circ$

Step3: Compare corresponding angles

Check for two matching angles.
$\triangle KML$ has angles $112^\circ, 40^\circ, 28^\circ$; $\triangle TUV$ has angles $112^\circ, 30^\circ, 38^\circ$. Only one pair of angles is equal.

Answer:

1. For problem 12: $x = 6$
2. For problem 13: $x = 38.6$
3. For problem 14: $\triangle TUV$ is not similar to $\triangle KML$. The only congruent angles are $\angle M = \angle U = 112^\circ$, but the other pairs of angles do not match, so the angle-angle similarity criterion is not satisfied.