Question

ting the side lengths and angle measures of one triangle given triangle ghj, the measure of angle g equals 110°, the measure of angle j equals 40°, and the measure of angle h equals 30°. complete the following statements. since angle g is \\(\boxed{}\\) angle, the opposite side, \\(\overline{jh}\\), is \\(\boxed{}\\). the order of the side lengths from longest to shortest is \\(\boxed{}\\).

Explanation:

Step1: Find measure of angle J

In a triangle, the sum of interior angles is \(180^\circ\). Given \(\angle G = 110^\circ\), \(\angle H = 30^\circ\), let \(\angle J=x\). Then \(110 + 30+x=180\), so \(x = 180-(110 + 30)=40^\circ\). So \(\angle J = 40^\circ\).

Step2: Analyze angle - side relationship

In a triangle, the larger the angle, the longer the side opposite to it. \(\angle G = 110^\circ\) (largest), \(\angle J = 40^\circ\), \(\angle H = 30^\circ\) (smallest). The sides opposite: \(\angle G\) opposite \(JH\), \(\angle J\) opposite \(GH\), \(\angle H\) opposite \(GJ\). Wait, correction: In triangle \(GHJ\), side opposite \(\angle G\) is \(JH\), opposite \(\angle H\) is \(GJ\), opposite \(\angle J\) is \(GH\). Since \(\angle G\) is the largest angle (\(110^\circ\)), the side opposite (\(JH\)) is the longest? Wait no, wait: Wait, angle \(G\) is at vertex \(G\), so sides: \(GJ\), \(GH\), \(JH\). Let's label the triangle: vertices \(G\), \(H\), \(J\). So side opposite \(\angle G\) (at \(G\)) is \(JH\), side opposite \(\angle H\) (at \(H\)) is \(GJ\), side opposite \(\angle J\) (at \(J\)) is \(GH\). Now, angles: \(\angle G = 110^\circ\), \(\angle J = 40^\circ\), \(\angle H = 30^\circ\). So order of angles from largest to smallest: \(\angle G>\angle J>\angle H\). Therefore, order of opposite sides from longest to shortest: \(JH\) (opposite \(\angle G\))? Wait no, wait \(\angle G\) is \(110^\circ\), so side opposite is \(JH\). \(\angle J\) is \(40^\circ\), opposite is \(GH\). \(\angle H\) is \(30^\circ\), opposite is \(GJ\). So since \(\angle G\) is largest, \(JH\) (opposite) is longest? Wait no, wait, no: Wait, angle at \(G\) is between sides \(GJ\) and \(GH\), so side opposite angle \(G\) is \(JH\). Angle at \(H\) is between \(GH\) and \(JH\), so side opposite is \(GJ\). Angle at \(J\) is between \(GJ\) and \(JH\), so side opposite is \(GH\). So angles: \(\angle G = 110^\circ\) (largest), \(\angle J = 40^\circ\), \(\angle H = 30^\circ\) (smallest). So sides opposite: \(JH\) (opp \(\angle G\)), \(GH\) (opp \(\angle J\)), \(GJ\) (opp \(\angle H\)). So order of angles: \(\angle G>\angle J>\angle H\), so order of opposite sides: \(JH>GH>GJ\). Wait, but the first statement: "Since angle \(G\) is [blank] angle, the opposite side, \(JH\), is [blank]". Since \(\angle G = 110^\circ\) is the largest angle, so "Since angle \(G\) is the largest angle, the opposite side, \(JH\), is the longest". Then order of side lengths from longest to shortest: \(JH\) (longest), then \(GH\) (opp \(\angle J = 40^\circ\)), then \(GJ\) (opp \(\angle H = 30^\circ\)). Wait, but let's re - check angle measures: \(\angle G = 110\), \(\angle J = 40\), \(\angle H = 30\). So angle sizes: \(110>40>30\), so sides opposite: \(JH\) (opp \(110\)), \(GH\) (opp \(40\)), \(GJ\) (opp \(30\)). So \(JH>GH>GJ\).

Wait, the first blank: "Since angle \(G\) is [the largest] angle, the opposite side, \(JH\), is [the longest]". Then order of side lengths from longest to shortest: \(JH\), \(GH\), \(GJ\) (or need to check labels again). Wait, maybe the triangle is labeled as \(G\), \(H\), \(J\) with \(G\) connected to \(J\) and \(G\) connected to \(H\), \(J\) connected to \(H\). So sides: \(GJ\), \(GH\), \(JH\). So side \(JH\) is between \(J\) and \(H\), \(GJ\) between \(G\) and \(J\), \(GH\) between \(G\) and \(H\). So angle at \(G\) is between \(GJ\) and \(GH\), so opposite side is \(JH\). Angle at \(J\) is between \(GJ\) and \(JH\), opposite side is \(GH\). Angle at \(H\) is between \(GH\) and \(JH\), opposite side is \(GJ\). So with angles: \(\angle G = 110\) (largest), \(\angle J = 40\), \(\angle H = 30\) (smallest). So sides: \(JH\) (opp \(\angle G\)) is longest, \(GH\) (opp \(\angle J\)) is middle, \(GJ\) (opp \(\angle H\)) is shortest. So order from longest to shortest: \(JH\), \(GH\), \(GJ\).

Answer:

First, measure of angle \(J\) is \(40^\circ\). Then, since angle \(G\) is the largest angle (\(110^\circ\)), the opposite side \(JH\) is the longest. The order of side lengths from longest to shortest is \(JH\), \(GH\), \(GJ\) (assuming standard triangle labeling). But based on the steps:

1. Measure of angle \(J\): \(40^\circ\) (since \(110 + 30+40 = 180\)).
2. Since angle \(G\) is the largest angle, the opposite side \(JH\) is the longest.
3. Order of side lengths from longest to shortest: \(JH\), \(GH\), \(GJ\) (or depending on exact side - angle correspondence, but the key is using the angle - side relationship: larger angle \(\implies\) longer opposite side).