Question

angle side relationship
question
in \\( \delta \mathrm{vwx}, \mathrm{m} \angle \mathrm{v}=102^{\circ} \\) and \\( \mathrm{m} \angle \mathrm{w}=51^{\circ} \\). which list
has the sides of \\( \delta \mathrm{vwx} \\) in order from shortest to
longest?
answer
\\( \bigcirc \mathrm{xv}, \mathrm{vw}, \mathrm{wx} \\)
\\( \bigcirc \mathrm{xv}, \mathrm{wx}, \mathrm{vw} \\)
\\( \bigcirc \mathrm{vw}, \mathrm{wx}, \mathrm{xv} \\)
\\( \bigcirc \mathrm{wx}, \mathrm{xv}, \mathrm{vw} \\)
\\( \bigcirc \mathrm{wx}, \mathrm{vw}, \mathrm{xv} \\)
\\( \bigcirc \mathrm{vw}, \mathrm{xv}, \mathrm{wx} \\)

Explanation:

Step1: Find the measure of ∠X

In a triangle, the sum of interior angles is \(180^\circ\). So, \(m\angle X=180^\circ - m\angle V - m\angle W\).
Substitute \(m\angle V = 102^\circ\) and \(m\angle W = 51^\circ\):
\(m\angle X=180^\circ - 102^\circ - 51^\circ = 27^\circ\)

Step2: Order the angles from smallest to largest

The angles are \(m\angle X = 27^\circ\), \(m\angle W = 51^\circ\), \(m\angle V = 102^\circ\). So, \(m\angle X

Step3: Relate angles to their opposite sides

In a triangle, the larger the angle, the longer its opposite side.

• Opposite of \(\angle X\) is \(VW\)
• Opposite of \(\angle W\) is \(XV\)
• Opposite of \(\angle V\) is \(WX\)

Since \(m\angle X

Wait, in \(\triangle VWX\), the vertices are V, W, X. So:

• Side opposite \(\angle V\) (vertex V) is \(WX\) (connecting W and X)
• Side opposite \(\angle W\) (vertex W) is \(VX\) (wait, the side labels: the triangle is \(\triangle VWX\), so the sides are:

• \(VW\): between V and W
• \(WX\): between W and X
• \(XV\): between X and V

So, angle at V: \(\angle V\), opposite side is \(WX\) (since it's opposite vertex V, so side WX)

Angle at W: \(\angle W\), opposite side is \(XV\) (opposite vertex W, so side XV)

Angle at X: \(\angle X\), opposite side is \(VW\) (opposite vertex X, so side VW)

So angles: \(\angle X = 27^\circ\) (opposite \(VW\)), \(\angle W = 51^\circ\) (opposite \(XV\)), \(\angle V = 102^\circ\) (opposite \(WX\))

So order of angles (smallest to largest): \(\angle X<\angle W<\angle V\)

Therefore, order of opposite sides (shortest to longest): \(VW\) (opposite \(\angle X\)), \(XV\) (opposite \(\angle W\)), \(WX\) (opposite \(\angle V\))? Wait, no, that can't be. Wait, no: the larger the angle, the longer the opposite side. So if angle X is smallest (27), its opposite side (VW) is shortest. Then angle W (51) is next, opposite side (XV) is next. Then angle V (102) is largest, opposite side (WX) is longest. So the sides from shortest to longest: \(VW\) (opposite 27), \(XV\) (opposite 51), \(WX\) (opposite 102). Wait, but let's check the options. Wait the options are:

Options:

1. XV, VW, WX

1. XV, WX, VW

1. VW, WX, XV

1. WX, XV, VW

1. WX, VW, XV

1. VW, XV, WX

Wait, let's re-express:

Angles:

\(\angle X = 27^\circ\) (opposite \(VW\))

\(\angle W = 51^\circ\) (opposite \(XV\))

\(\angle V = 102^\circ\) (opposite \(WX\))

So the sides opposite these angles:

• \(VW\) (opposite \(\angle X\), 27°) → shortest

• \(XV\) (opposite \(\angle W\), 51°) → middle

• \(WX\) (opposite \(\angle V\), 102°) → longest

So the order from shortest to longest is \(VW\), \(XV\), \(WX\), which is option 6: VW, XV, WX.

Wait, let's confirm again:

In triangle, side opposite smaller angle is shorter.

Angle X: 27°, opposite side VW → shortest.

Angle W: 51°, opposite side XV → next.

Angle V: 102°, opposite side WX → longest.

So the sides from shortest to longest: VW, XV, WX.

Answer:

VW, XV, WX (the sixth option: VW, XV, WX)