Question

in the figure, m∠dbc = m∠cad, m∠dba = m∠cab, and ec = ed. if m∠cba = 5x + 2, ec = 3, and eb = 8, find the following values. 29. m∠dbc 30. m∠cba 31. m∠cab 29. m∠dbc =

Explanation:

Step1: Use the property of equal - length sides

Since \(EC = ED = 3\), we are not directly using this information for angle - related calculations for now. But we focus on the angle - equality relationships.

Step2: Use the angle - equality to set up an equation for \(x\)

Because \(\angle DBA=\angle CAB\), we have \(3x - 8=2x + 1\).
Solve for \(x\):

$$
LATEXBLOCK0
$$

Step3: Find \(m\angle CBA\)

We know that \(m\angle CBA=5x + 2\). Substitute \(x = 9\) into the formula:

$$m\angle CBA=5\times9+2=45 + 2=47^{\circ}$$

Step4: Find \(m\angle CAB\)

Since \(m\angle CAB = 2x+1\) and \(x = 9\), then \(m\angle CAB=2\times9 + 1=18+1 = 19^{\circ}\)

Step5: Find \(m\angle DBC\)

We know that \(m\angle DBC=m\angle CAD\) and \(m\angle CBA=m\angle DBA + m\angle DBC\). Also, \(m\angle DBA=m\angle CAB\).
Since \(m\angle CBA = 47^{\circ}\) and \(m\angle DBA=m\angle CAB = 19^{\circ}\), then \(m\angle DBC=m\angle CBA-m\angle DBA=47^{\circ}-19^{\circ}=28^{\circ}\)

Answer:

1. \(28\)
2. \(47\)
3. \(19\)