Question

1. question 7. in a right triangle, if one of the acute angles measures 30 degrees, the other acute angle will measure:

a) 30 degrees.
b) 45 degrees.
c) 60 degrees.
d) 90 degrees.

1. question 8. the cosine of an angle in a right triangle represents:

a) the length of the side opposite the angle.
b) the length of the side adjacent to the angle.
c) the length of the hypotenuse.
d) the area of the triangle.

1. question 9. if the sine of an angle in a right triangle is 0.5, the cosine of the complementary angle is:

a) 0
b) 0.5
c) 1
d) 2

Explanation:

Step1: Sum of triangle angles

The total interior angles of any triangle equal $180^\circ$. A right triangle has one $90^\circ$ angle.

Step2: Calculate unknown acute angle

Let the unknown acute angle be $x$. Set up the equation:
$$90^\circ + 30^\circ + x = 180^\circ$$
Solve for $x$:
$$x = 180^\circ - 90^\circ - 30^\circ = 60^\circ$$

Step3: Recall cosine definition

In a right triangle, cosine of an acute angle is defined as the ratio of the adjacent side length to the hypotenuse length. Among the options, only option b matches the component of this definition related to side length.

Step4: Use complementary angle identity

For complementary angles $\theta$ and $90^\circ-\theta$, the identity $\sin\theta = \cos(90^\circ-\theta)$ holds. Given $\sin\theta=0.5$, so $\cos(90^\circ-\theta)=0.5$.

Answer:

1. c) 60 degrees.
2. b) The length of the side adjacent to the angle.
3. b) 0.5