Question

which information do you know is true from the diagram? check all that apply

□ ∠abd is a right angle

□ (overrightarrow{bc}) bisects angle ∠abd

□ (mangle abc=45^{circ})

□ (mangle cbd = 45^{circ})

□ (mangle abd=90^{circ})

Explanation:

Step1: Identify right - angle

The small square at ∠ABD indicates it is a right angle. So \(m\angle ABD=90^{\circ}\) (by the definition of a right - angle).

Step2: Use angle - bisector property

Since \(\overrightarrow{BC}\) bisects ∠ABD (given by the congruent angle markings), and \(m\angle ABD = 90^{\circ}\).

Step3: Calculate \(m\angle ABC\) and \(m\angle CBD\)

By the angle - bisector theorem, if an angle is bisected, the two resulting angles are equal. Let \(m\angle ABC=x\) and \(m\angle CBD = y\). We know \(x + y=m\angle ABD\), and \(x = y\) (because \(\overrightarrow{BC}\) bisects ∠ABD). Substituting \(m\angle ABD = 90^{\circ}\) into \(x + y=90^{\circ}\) and \(x = y\), we get \(2x=90^{\circ}\), so \(x=y = 45^{\circ}\). That is \(m\angle ABC=45^{\circ}\) and \(m\angle CBD = 45^{\circ}\).

Answer:

• ∠ABD is a right angle
• \(\overrightarrow{BC}\) bisects angle ∠ABD
• \(m\angle ABC = 45^{\circ}\)
• \(m\angle CBD = 45^{\circ}\)
• \(m\angle ABD = 90^{\circ}\)