Question

make connections solve the problem. in triangle rst, m∠t is 5° more than m∠r, and m∠s is 10° less than m∠t. what is the measure of each angle? m∠r =, m∠s =, m∠t =

Explanation:

Step1: Define variables

Let \( m\angle R = x \). Then \( m\angle T = x + 5 \) (since \( m\angle T \) is \( 5^\circ \) more than \( m\angle R \)), and \( m\angle S = (x + 5)- 10=x - 5 \) (since \( m\angle S \) is \( 10^\circ \) less than \( m\angle T \)).

Step2: Use triangle angle sum theorem

The sum of the interior angles of a triangle is \( 180^\circ \). So, \( m\angle R+m\angle S + m\angle T=180^\circ \). Substituting the expressions for each angle:

$$ x+(x - 5)+(x + 5)=180 $$

Step3: Simplify and solve for \( x \)

Simplify the left - hand side of the equation:

$$ x+x - 5+x + 5=180 $$

The \(-5\) and \(+5\) cancel out, leaving \( 3x=180 \). Divide both sides by 3: \( x=\frac{180}{3}=60 \).

Step4: Find the measures of each angle

• For \( m\angle R \): Since \( m\angle R=x \), \( m\angle R = 60^\circ \).
• For \( m\angle T \): \( m\angle T=x + 5=60 + 5=65^\circ \).
• For \( m\angle S \): \( m\angle S=x - 5=60 - 5=55^\circ \).

Answer:

\( m\angle R=\boldsymbol{60^\circ} \), \( m\angle S=\boldsymbol{55^\circ} \), \( m\angle T=\boldsymbol{65^\circ} \)