Question

list the side lengths from shortest to longest for each diagram.
a.
b.

Explanation:

Step1: Find the third angle in triangle \(LMK\)

The sum of angles in a triangle is \(180^{\circ}\). Let \(\angle M = 180-(28 + 118)=34^{\circ}\).
In a triangle, the side opposite the smaller angle is shorter.
\(\angle M=34^{\circ}\), \(\angle L = 28^{\circ}\), \(\angle K=118^{\circ}\)
Side opposite \(\angle L\) is \(l\), side opposite \(\angle M\) is \(m\), side opposite \(\angle K\) is \(k\)
Since \(28^{\circ}<34^{\circ}<118^{\circ}\), then \(l < m

Step2: Analyze triangle \(XWY\)

In \(\triangle XWY\), \(\angle X = 27^{\circ}\), \(\angle XYW=67^{\circ}\), then \(\angle XWY=180-(27 + 67)=86^{\circ}\)
Side opposite \(\angle X\) is \(c\), side opposite \(\angle XYW\) is \(a\), side opposite \(\angle XWY\) is \(b\)
Since \(27^{\circ}<67^{\circ}<86^{\circ}\), then \(c < a < b\)

Step3: Analyze triangle \(YWZ\)

In \(\triangle YWZ\), \(\angle YWZ = 64^{\circ}\), \(\angle YZW=180-(64 + 79)=37^{\circ}\)
Side opposite \(\angle YZW\) is \(c\), side opposite \(\angle YWZ\) is \(d\), side opposite \(\angle WYZ\) is \(e\)
Since \(37^{\circ}<64^{\circ}<79^{\circ}\), then \(c

Combining for part \(b\):
In \(\triangle XWZ\) (considering the two - part triangle), we know from angle - side relationships
For the whole figure in part \(b\):
In \(\triangle XWY\), angles: \(\angle X = 27^{\circ}\), \(\angle XYW = 67^{\circ}\), \(\angle XWY=86^{\circ}\)
In \(\triangle YWZ\), angles: \(\angle YZW = 37^{\circ}\), \(\angle YWZ = 64^{\circ}\), \(\angle WYZ=79^{\circ}\)
The sides:
The side opposite \(27^{\circ}\) is \(c\), side opposite \(37^{\circ}\) is \(d\), side opposite \(64^{\circ}\) is \(e\), side opposite \(67^{\circ}\) is \(a\), side opposite \(79^{\circ}\) is \(b\) (after re - checking angle - side correspondences)
\(c < a

Answer:

a. \(l < mb. \(c < a