Question

in the figure, $m\angle 1 = (5x)\degree$ and $m\angle 2 = (x - 6)\degree$.
(a) write an equation to find $x$. make sure you use an \=\ sign in your answer.
equation:
(b) find the degree measure of each angle.
$m\angle 1 = \square\degree$
$m\angle 2 = \square\degree$

Explanation:

Part (a)

Step1: Identify angle relationship

From the figure, ∠1 and ∠2 form a right angle (90°), so their measures sum to 90°.

Step2: Write the equation

Given \( m\angle1 = (5x)^\circ \) and \( m\angle2 = (x - 6)^\circ \), the equation is \( 5x+(x - 6)=90 \). Simplifying the left - hand side, we get \( 5x+x - 6=90 \), which further simplifies to \( 6x-6 = 90 \), and then \( 5x+(x - 6)=90 \) can also be written as \( 5x+(x - 6)=90 \) or \( 6x-6 = 90 \). But the most direct equation from the angle sum is \( 5x+(x - 6)=90 \), and if we simplify the left side: \( 5x+x-6=6x - 6\), so the equation is \( 6x-6 = 90 \) or \( 5x+(x - 6)=90 \). However, the standard way is to use the sum of the two angles equal to 90 degrees. So the equation is \( 5x+(x - 6)=90 \) (or simplified to \( 6x-6 = 90 \)). But let's solve it step - by - step properly.
First, since the two angles form a right angle, \( m\angle1+m\angle2 = 90^\circ \). Substitute the given expressions: \( 5x+(x - 6)=90 \).

Step1: Solve the equation for x

We have the equation \( 5x+(x - 6)=90 \).
First, combine like terms: \( 5x+x-6=90\Rightarrow6x - 6=90 \).
Then, add 6 to both sides of the equation: \( 6x-6 + 6=90 + 6\Rightarrow6x=96 \).
Next, divide both sides by 6: \( x=\frac{96}{6}=16 \).

Step2: Find \( m\angle1 \)

Substitute \( x = 16 \) into the expression for \( m\angle1 \): \( m\angle1=5x=5\times16 = 80^\circ \).

Step3: Find \( m\angle2 \)

Substitute \( x = 16 \) into the expression for \( m\angle2 \): \( m\angle2=x - 6=16 - 6 = 10^\circ \).

Answer:

\( 5x+(x - 6)=90 \) (or \( 6x-6 = 90 \))

Part (b)