Question

in $\triangle opq$, $m\angle o = 19\degree$ and $m\angle p = 71\degree$. which list has the sides of $\triangle opq$ in order from shortest to longest?
shortest choose
choose
longest choose

Explanation:

Step1: Find the measure of ∠Q

In a triangle, the sum of interior angles is \(180^\circ\). So, \(m\angle Q=180^\circ - m\angle O - m\angle P\). Substituting \(m\angle O = 19^\circ\) and \(m\angle P = 71^\circ\), we get \(m\angle Q=180^\circ - 19^\circ - 71^\circ=90^\circ\).

Step2: Relate angles to opposite sides

In a triangle, the larger the angle, the longer the side opposite to it. The sides opposite to \(\angle O\), \(\angle P\), and \(\angle Q\) are \(PQ\), \(OQ\), and \(OP\) respectively.

We have \(m\angle O = 19^\circ\), \(m\angle P = 71^\circ\), \(m\angle Q = 90^\circ\). So, \(19^\circ<71^\circ<90^\circ\), which means \(m\angle O < m\angle P < m\angle Q\). Therefore, the sides opposite these angles will follow the same order: \(PQ < OQ < OP\) (since side opposite \(\angle O\) is \(PQ\), opposite \(\angle P\) is \(OQ\), opposite \(\angle Q\) is \(OP\)).

Answer:

Shortest: \(PQ\)
Middle: \(OQ\)
Longest: \(OP\)