Question

design camila is designing a new image for her cell phone case. if ( mangle abc = 35^{circ}), ( mangle bac = 29^{circ}), and ( angle acbcongangle deb), what is ( mangle deb)?

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). In \(\triangle ABC\), \(m\angle ABC + m\angle BAC + m\angle ACB = 180^\circ\).

Step2: Substitute known angles

We know \(m\angle ABC = 35^\circ\) and \(m\angle BAC = 29^\circ\). Substitute these values: \(35^\circ + 29^\circ + m\angle ACB = 180^\circ\).

Step3: Calculate \(m\angle ACB\)

First, add \(35^\circ\) and \(29^\circ\): \(35 + 29 = 64\). Then, \(m\angle ACB = 180^\circ - 64^\circ = 116^\circ\)? Wait, no, wait. Wait, \(35 + 29 = 64\), so \(m\angle ACB = 180 - 64 = 116\)? Wait, no, that can't be. Wait, no, wait, \(35 + 29 = 64\), so \(180 - 64 = 116\)? Wait, no, maybe I made a mistake. Wait, no, \(35 + 29 = 64\), so \(180 - 64 = 116\)? Wait, but the problem says \(\angle ACB \cong \angle DEB\). Wait, no, wait, maybe I misread the angles. Wait, \(\angle ABC = 35^\circ\), \(\angle BAC = 29^\circ\), so in \(\triangle ABC\), angles sum to 180. So \(35 + 29 + \angle ACB = 180\). So \(64 + \angle ACB = 180\), so \(\angle ACB = 180 - 64 = 116\)? Wait, that seems large, but maybe. Then, since \(\angle ACB \cong \angle DEB\), so \(m\angle DEB = m\angle ACB = 116^\circ\)? Wait, no, wait, maybe I messed up the triangle. Wait, maybe \(\triangle ABC\) is a triangle with angles at A, B, C. So angle at B is 35, angle at A is 29, so angle at C is 180 - 35 - 29 = 116. Then angle DEB is congruent to angle ACB, so DEB is 116? Wait, but that seems big. Wait, maybe I made a mistake. Wait, no, 35 + 29 is 64, 180 - 64 is 116. So yes, that's correct. Then, since \(\angle ACB \cong \angle DEB\), so \(m\angle DEB = 116^\circ\)? Wait, but maybe the triangle is different. Wait, maybe the diagram shows that \(\triangle ABC\) is a triangle, and \(\angle ACB\) is congruent to \(\angle DEB\). So yes, the calculation is 180 - 35 - 29 = 116. So \(m\angle DEB = 116^\circ\). Wait, but let me check again. 35 + 29 = 64, 180 - 64 = 116. Yes. So that's the measure.

Answer:

\(116^\circ\)