Question

given: \\( \overrightarrow{ad} \\).
\\( \overrightarrow{eb} \\) bisects \\( \angle aec \\)
what statements are true regarding the given statement
and diagram?
\\( \square \angle ced \\) is a right angle.
\\( \square \angle cea \\) is a right angle.
\\( \square m \angle cea=\frac{1}{2}(m \angle ceb) \\)
\\( \square m \angle ceb=m \angle bea \\)
\\( \square m \angle deb=135^{\circ} \\)
\\( \square m \angle aeb=35^{\circ} \\)

Explanation:

Step1: Analyze ∠CED and ∠CEA

From the diagram, ∠CED and ∠CEA are right angles (since there's a right - angle symbol at E between EC and ED, and AD is a straight line, so ∠CEA is also a right angle as ∠AED is a straight angle and ∠CED = 90°). So ∠CED is a right angle (True), ∠CEA is a right angle (True).

Step2: Analyze \(m\angle CEA=\frac{1}{2}(m\angle CEB)\)

Since EB bisects ∠AEC and ∠AEC = 90°, then ∠AEB=∠BEC = 45°. So \(m\angle CEA = 90^{\circ}\), \(m\angle CEB=45^{\circ}\), and \(90^{\circ}=\frac{1}{2}(m\angle CEB)\) is false (it should be \(m\angle CEB=\frac{1}{2}(m\angle CEA)\)).

Step3: Analyze \(m\angle CEB = m\angle BEA\)

Since EB bisects ∠AEC, ∠AEB and ∠BEC are equal. So \(m\angle CEB=m\angle BEA\) (True).

Step4: Analyze \(m\angle DEB\)

∠CED = 90°, ∠CEB = 45°, so \(m\angle DEB=\angle CED+\angle CEB=90^{\circ} + 45^{\circ}=135^{\circ}\) (True).

Step5: Analyze \(m\angle AEB\)

Since EB bisects ∠AEC (\(m\angle AEC = 90^{\circ}\)), \(m\angle AEB=\frac{1}{2}\times90^{\circ} = 45^{\circ}
eq35^{\circ}\), so this is false.

Answer:

• ∠CED is a right angle.
• ∠CEA is a right angle.
• \(m\angle CEB = m\angle BEA\)
• \(m\angle DEB = 135^{\circ}\)