Question

in $\triangle ijk$, $m\angle i = 30^\circ$ and $m\angle j = 89^\circ$. which statement about the sides of $\triangle ijk$ must be true?
answer
\\(\circ\\) $ij < ki < jk$ \\(\circ\\) $ki < ij < jk$
\\(\circ\\) $jk < ij < ki$ \\(\circ\\) $ij < jk < ki$
\\(\circ\\) $ki < jk < ij$ \\(\circ\\) $jk < ki < ij$

Explanation:

Step1: Find $\angle K$

The sum of angles in a triangle is $180^\circ$.
$\angle K = 180^\circ - 30^\circ - 89^\circ = 61^\circ$

Step2: Match sides to opposite angles

In a triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle:

• $\angle I = 30^\circ$ (smallest) $

ightarrow$ opposite side $JK$ (shortest)

• $\angle K = 61^\circ$ (middle) $

ightarrow$ opposite side $IJ$ (middle length)

• $\angle J = 89^\circ$ (largest) $

ightarrow$ opposite side $KI$ (longest)

Step3: Order the sides

Arrange sides from shortest to longest: $JK < IJ < KI$

Answer:

JK < IJ < KI