Question

1. given m∠cba = 90°. claudia claims that ∠abd and ∠cbd must have the same measure. erika claims that ∠abd and ∠cbd must be acute angles. (a) what tools could you use to determine whether claudia is right or wrong? is claudia correct? (b) is erika correct? why?

Explanation:

Step1: Recall angle - related concepts

An acute angle measures less than 90°. We know that \(\angle CBA=90^{\circ}\). If \(\angle ABD+\angle CBD = \angle CBA\).

Step2: Analyze Claudia's claim

Claudia claims that \(\angle ABD\) and \(\angle CBD\) must have the same measure. This is wrong. For example, \(\angle ABD = 30^{\circ}\) and \(\angle CBD=60^{\circ}\), and \(30^{\circ}+60^{\circ}=90^{\circ}\). They do not have to be equal.

Step3: Analyze Erika's claim

Erika claims that \(\angle ABD\) and \(\angle CBD\) must be acute angles. Since \(\angle CBA = 90^{\circ}\) and \(\angle ABD+\angle CBD=\angle CBA\), and an acute - angle is less than \(90^{\circ}\), both \(\angle ABD\) and \(\angle CBD\) must be less than \(90^{\circ}\), so Erika is correct.

Step4: Tools to use

To determine if Claudia is right or wrong, we can use a protractor to measure the angles \(\angle ABD\) and \(\angle CBD\) and check if they are equal. To determine if Erika is right, we can also use a protractor to measure the angles and check if they are less than \(90^{\circ}\).

Answer:

(a) Tools: Protractor. Claudia is wrong because \(\angle ABD\) and \(\angle CBD\) do not have to have the same measure as long as their sum is \(90^{\circ}\).
(b) Erika is correct because since \(\angle CBA = 90^{\circ}\) and \(\angle ABD+\angle CBD=\angle CBA\), both \(\angle ABD\) and \(\angle CBD\) must be less than \(90^{\circ}\) (acute angles).