Question

if ( mangle ebd = 4x + 16 ) and ( mangle dbc = 6x + 4 ), find ( mangle ebd ).

Explanation:

Assuming that \(\angle EBD\) and \(\angle DBC\) are complementary (since the problem likely involves a right angle, as it's a common angle - relationship problem in geometry), so \(m\angle EBD + m\angle DBC=90^{\circ}\) (if they form a right angle) or supplementary (if they form a straight angle, \(180^{\circ}\)). But from the expressions, it's more likely a right angle (since the coefficients of \(x\) and constants suggest a right - angle sum). So we set up the equation:

Step 1: Set up the equation

Since \(\angle EBD\) and \(\angle DBC\) are complementary (assuming a right - angle situation, so their measures add up to \(90^{\circ}\)), we have \((4x + 16)+(6x + 4)=90\).
Simplify the left - hand side: \(4x+16 + 6x+4=10x + 20\). So the equation becomes \(10x+20 = 90\).

Step 2: Solve for \(x\)

Subtract 20 from both sides of the equation: \(10x+20-20=90 - 20\), which gives \(10x=70\).
Divide both sides by 10: \(x=\frac{70}{10}=7\).

Step 3: Find \(m\angle EBD\)

Substitute \(x = 7\) into the expression for \(m\angle EBD\), which is \(4x + 16\).
\(m\angle EBD=4\times7+16=28 + 16=44\).

(If the angles were supplementary, \((4x + 16)+(6x + 4)=180\), \(10x+20 = 180\), \(10x=160\), \(x = 16\), and \(m\angle EBD=4\times16 + 16=64 + 16 = 80\). But the first case is more likely for a typical geometry problem with these angle expressions. Since the problem is about angle measures in geometry, we assume the right - angle case here.)

Answer:

\(44^{\circ}\) (If supplementary, the answer would be \(80^{\circ}\), but based on the common problem structure, \(44^{\circ}\) is more likely. If there was a diagram indicating a right angle, this is correct. If it was a straight angle, the other answer holds. Since the diagram is not fully shown, but the expressions suggest a right - angle problem, we go with \(44^{\circ}\))