Question

reasoning solve the problem. in triangle rst, ( mangle t ) is ( 5^{circ} ) more than ( mangle r ), and ( mangle s ) is ( 10^{circ} ) less than ( mangle t ). what is the measure of each angle? ( mangle r=), ( mangle s=), ( mangle t=)

Explanation:

Step1: Express angles in terms of \( \angle R \)

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 the triangle - angle sum theorem

The sum of the interior angles of a triangle is \( 180^{\circ} \). So \( m\angle R+m\angle S+m\angle T = 180^{\circ} \).
Substitute the expressions: \( x+(x - 5)+(x + 5)=180 \).
Simplify the left - hand side: \( x+x-5+x + 5=3x \).
So \( 3x=180 \).
Solve for \( x \): \( x=\frac{180}{3}=60 \).

Step3: Find the measures of each angle

Since \( x = m\angle R \), then \( m\angle R = 60^{\circ} \).
\( m\angle T=x + 5=60+5 = 65^{\circ} \).
\( m\angle S=x-5=60 - 5=55^{\circ} \).

Answer:

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