Question

1. in \\( \triangle rst \\), if \\( m\angle r \\) is five more than twice \\( x \\), \\( m\angle s \\) is one more than \\( x \\), and \\( m\angle t \\) is sixteen less than seven times \\( x \\), find \\( x \\) and the measure of each angle.\\( x = \underline{\quad\quad} \\)\\( m\angle r = \underline{\quad\quad} \\)\\( m\angle s = \underline{\quad\quad} \\)\\( m\angle t = \underline{\quad\quad} \\)15. in \\( \triangle abc \\), if \\( m\angle a \\) is thirteen less than \\( m\angle c \\) and \\( m\angle b \\) is eleven less than four times \\( m\angle c \\), find the measure of each angle.\\( m\angle a = \underline{\quad\quad} \\)\\( m\angle b = \underline{\quad\quad} \\)\\( m\angle c = \underline{\quad\quad} \\)

Explanation:

Problem 14:

Step1: Define variables and angle sum

Let \( x \) be a value. \( m\angle R = 2x + 5 \), \( m\angle S = x + 1 \), \( m\angle T = 7x - 16 \). The sum of angles in a triangle is \( 180^\circ \), so \( (2x + 5) + (x + 1) + (7x - 16) = 180 \).

Step2: Solve for \( x \)

Combine like terms: \( 2x + x + 7x + 5 + 1 - 16 = 180 \) → \( 10x - 10 = 180 \). Add 10: \( 10x = 190 \) → \( x = 19 \).

Step3: Find each angle

\( m\angle R = 2(19) + 5 = 43^\circ \), \( m\angle S = 19 + 1 = 20^\circ \), \( m\angle T = 7(19) - 16 = 117^\circ \).

Step1: Define variables and angle sum

Let \( m\angle C = x \). Then \( m\angle A = x - 13 \), \( m\angle B = 4x - 11 \). Sum of angles in a triangle: \( (x - 13) + (4x - 11) + x = 180 \).

Step2: Solve for \( x \)

Combine like terms: \( x + 4x + x - 13 - 11 = 180 \) → \( 6x - 24 = 180 \). Add 24: \( 6x = 204 \) → \( x = 34 \).

Step3: Find each angle

\( m\angle A = 34 - 13 = 21^\circ \), \( m\angle B = 4(34) - 11 = 125^\circ \), \( m\angle C = 34^\circ \).

Answer:

\( x = 19 \), \( m\angle R = 43^\circ \), \( m\angle S = 20^\circ \), \( m\angle T = 117^\circ \)

Problem 15: