Question

1. use the diagram to the right to answer the following questions. m∠1 = ____ m∠2 = __ m∠3 = __ m∠4 = __ 23. use the diagram to the right to find x. x = ____

Explanation:

Step1: Find $m\angle1$ in first triangle

In the first - triangle, using the fact that the sum of angles in a right - triangle is 180°. One angle is 90° and another is 20°. So $m\angle1=180^{\circ}-90^{\circ}-20^{\circ}=70^{\circ}$.

Step2: Find $m\angle2$

Since $\angle1$ and $\angle2$ are complementary (they form a right - angle), $m\angle2 = 90^{\circ}-m\angle1=90^{\circ}-70^{\circ}=20^{\circ}$.

Step3: Find $m\angle3$

In the right - triangle with $\angle3$, one angle is 90° and another is 63°. So $m\angle3=180^{\circ}-90^{\circ}-63^{\circ}=27^{\circ}$.

Step4: Find $m\angle4$

The sum of angles in the large triangle is 180°. We know two of the non - $\angle4$ angles from previous steps. $m\angle4=180^{\circ}-20^{\circ}-(90^{\circ}+27^{\circ})=43^{\circ}$.

Step5: Solve for $x$ in second triangle

The sum of the interior angles of a triangle is 180°. So $(4x - 6)+(2x + 13)+(3x-13)=180$.
Combining like terms: $4x+2x + 3x-6 + 13-13=180$, which simplifies to $9x-6 = 180$.
Adding 6 to both sides: $9x=180 + 6=186$.
Dividing both sides by 9: $x=\frac{186}{9}=\frac{62}{3}\approx20.67$.

Answer:

$m\angle1 = 70^{\circ}$
$m\angle2 = 20^{\circ}$
$m\angle3 = 27^{\circ}$
$m\angle4 = 43^{\circ}$
$x=\frac{62}{3}$