Question

a. the greatest angle of △abc has a measure of 104°, and the other angle measures are 30° and 46°. the smallest angle is opposite the shortest side. ∠a measures __ °. ∠b measures °. ∠c measures __ °.

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). Given the greatest angle is \(104^\circ\), and the other two are \(30^\circ\) and \(46^\circ\). From the diagram, \(\angle A\) appears to be the largest angle (as it's the top angle in the triangle with base \(BC\), making it obtuse), so \(\angle A = 104^\circ\).

Step2: Identify smallest angle

The smallest angle among \(30^\circ\) and \(46^\circ\) is \(30^\circ\). In a triangle, the smallest angle is opposite the shortest side. Looking at the triangle, \(\angle B\) is the smallest angle (visually, as \(BC\) is the base and \(B\) is a smaller angle), so \(\angle B = 30^\circ\).

Step3: Calculate remaining angle

Using the angle - sum property: \(\angle A+\angle B+\angle C = 180^\circ\). Substitute \(\angle A = 104^\circ\) and \(\angle B = 30^\circ\) into the formula: \(104^\circ+30^\circ+\angle C=180^\circ\). Then \(\angle C=180^\circ-(104^\circ + 30^\circ)=180^\circ - 134^\circ = 46^\circ\).

Answer:

\(\angle A\) measures \(104^\circ\).
\(\angle B\) measures \(30^\circ\).
\(\angle C\) measures \(46^\circ\).