Question

a. determine the measure of an angle whose measure is 5 times that of its complement.
b. if two angles of a triangle are complementary, what is the measure of the third angle?

a. the angle measures \\(\square^\circ\\).

Explanation:

Part a

Step1: Define variables

Let the measure of the complement angle be \( x \) degrees. Then the angle we want to find is \( 5x \) degrees.

Step2: Use complementary angle property

Complementary angles add up to \( 90^\circ \), so we have the equation \( x + 5x=90 \).

Step3: Solve the equation

Combine like terms: \( 6x = 90 \). Then divide both sides by 6: \( x=\frac{90}{6}=15 \).

Step4: Find the angle

The angle is \( 5x \), so substitute \( x = 15 \): \( 5\times15 = 75 \).

Step1: Recall triangle angle sum

The sum of the interior angles of a triangle is \( 180^\circ \).

Step2: Use complementary angles

If two angles are complementary, their sum is \( 90^\circ \). Let the third angle be \( y \). Then \( 90 + y=180 \).

Step3: Solve for \( y \)

Subtract 90 from both sides: \( y = 180 - 90=90 \).

Answer:

\( 75 \)

Part b