Question

what is the measure of each angle? drag an angle measure into each box. not all angle measures will be used m∠dbe m∠deb m∠abe m∠def 35° 53° 92° 127° 145°

Explanation:

Step1: Use the exterior - angle property of a triangle

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. In \(\triangle BDE\), \(\angle BDE = 85^{\circ}\), \(\angle DBE=(4x + 5)^{\circ}\), and \(\angle DEB=(3x - 1)^{\circ}\). So, \(85=(4x + 5)+(3x - 1)\).

Step2: Simplify the equation

Combine like terms: \(85 = 4x+5 + 3x-1\), which simplifies to \(85=7x + 4\).

Step3: Solve for \(x\)

Subtract 4 from both sides of the equation: \(85−4 = 7x\), so \(81 = 7x\). Then \(x=\frac{81}{7}\approx11.57\).
Let's use another property. The sum of angles in a triangle is \(180^{\circ}\). Consider the linear pair with \(\angle BDE\). The adjacent angle to \(\angle BDE\) in the triangle is \(180 - 85=95^{\circ}\).
The sum of the other two angles in the triangle \(\triangle BDE\) is \(180 - 95 = 85^{\circ}\), so \((4x + 5)+(3x - 1)=85\).
Combining like terms gives \(7x+4 = 85\), then \(7x=81\), \(x = 12\).

Step4: Find the measure of each angle

• For \(\angle DBE=(4x + 5)^{\circ}\), substitute \(x = 12\): \(m\angle DBE=4\times12 + 5=48 + 5=53^{\circ}\).
• For \(\angle DEB=(3x - 1)^{\circ}\), substitute \(x = 12\): \(m\angle DEB=3\times12-1=36 - 1 = 35^{\circ}\).
• \(\angle ABE\) and \(\angle DBE\) form a linear pair. So \(m\angle ABE=180 - m\angle DBE=180 - 53 = 127^{\circ}\).
• \(\angle DEF\) and \(\angle DEB\) form a linear pair. So \(m\angle DEF=180 - m\angle DEB=180 - 35 = 145^{\circ}\).

Answer:

\(m\angle DBE = 53^{\circ}\)
\(m\angle DEB = 35^{\circ}\)
\(m\angle ABE = 127^{\circ}\)
\(m\angle DEF = 145^{\circ}\)