Question

triangle angle sum
solve for the given variable.
1)

b = ____
2)

r = ____
3)

z = ____
4)

a = ____
5)

e = ____
6)

s = ____
7)

d = ____
8)

k = ____

Explanation:

Problem 1: Solve for \( b \)

Step 1: Find the adjacent angle to \( 113^\circ \)

The angle adjacent to \( 113^\circ \) on a straight line is \( 180^\circ - 113^\circ = 67^\circ \).

Step 2: Use triangle angle sum (sum is \( 180^\circ \))

The angles in the triangle are \( 55^\circ \), \( 6b + 4^\circ \), and \( 67^\circ \). So:
\( 55 + (6b + 4) + 67 = 180 \)
Simplify: \( 6b + 126 = 180 \)
Subtract 126: \( 6b = 54 \)
Divide by 6: \( b = 9 \)

Step 1: Find the third angle in the triangle

The triangle has angles \( 84^\circ \), \( 48^\circ \), so the third angle is \( 180 - 84 - 48 = 48^\circ \).

Step 2: Use linear pair (sum to \( 180^\circ \))

The angle \( 6r^\circ \) and the third angle form a linear pair: \( 6r + 48 = 180 \)
Subtract 48: \( 6r = 132 \)
Divide by 6: \( r = 22 \)

Step 1: Find the third angle in the triangle

The angle adjacent to \( 66^\circ \) (linear pair) is \( 180 - 66 = 114^\circ \).
The triangle has angles \( 48^\circ \), \( 3z^\circ \), and \( 114^\circ \)? Wait, no—wait, the exterior angle is \( 66^\circ \), so the remote interior angles are \( 48^\circ \) and \( 3z^\circ \). By exterior angle theorem: \( 48 + 3z = 66 \)
Subtract 48: \( 3z = 18 \)
Divide by 3: \( z = 6 \)

Answer:

\( b = 9 \)

Problem 2: Solve for \( r \)