Question

determine whether the pair of figures is similar. if so, find the scale factor. explain your reasoning.

(image of two right triangles: triangle wzy with right angle at z, sides zw=5, zy=4, wy=6.4, angles at w: 39°, at y: 51°. triangle mkl with right angle at k, sides mk=8.2, kl=7.4, ml=11, angles at m: 42°, at l: 48°.)

Explanation:

Step1: Check Angles for Similarity

First, analyze the angles of both triangles. In triangle \( WZY \), we have a right angle at \( Z \) (\( 90^\circ \)), \( \angle W = 39^\circ \), so \( \angle Y = 180^\circ - 90^\circ - 39^\circ = 51^\circ \). In triangle \( MKL \), there is a right angle at \( K \) (\( 90^\circ \)), \( \angle M = 42^\circ \), so \( \angle L = 180^\circ - 90^\circ - 42^\circ = 48^\circ \)? Wait, no, wait. Wait, in triangle \( WZY \), angles are \( 90^\circ \), \( 39^\circ \), \( 51^\circ \). In triangle \( MKL \), angles: right angle at \( K \), \( \angle M = 42^\circ \), \( \angle L = 48^\circ \). Wait, that seems inconsistent? Wait, no, maybe I miscalculated. Wait, let's re - check triangle \( WZY \): \( \angle Z = 90^\circ \), \( \angle W = 39^\circ \), so \( \angle Y = 180 - 90 - 39 = 51^\circ \) (correct). Triangle \( MKL \): \( \angle K = 90^\circ \), \( \angle M = 42^\circ \), so \( \angle L = 180 - 90 - 42 = 48^\circ \). Wait, but in triangle \( WZY \), \( \angle Y = 51^\circ \), and in triangle \( MKL \), \( \angle L = 48^\circ \). Wait, maybe there's a mistake here. Wait, no, maybe the angle labels are different. Wait, let's check the sides. Wait, triangle \( WZY \): sides \( ZW = 5 \), \( ZY = 4 \), \( WY = 6.4 \). Triangle \( MKL \): sides \( MK = 8.2 \), \( KL = 7.4 \), \( ML = 11 \). Wait, maybe we should check the angle - angle (AA) similarity criterion. Let's find the angles correctly.

In triangle \( WZY \):
- \( \angle Z = 90^\circ \)
- \( \angle W = 39^\circ \)
- \( \angle Y = 51^\circ \)

In triangle \( MKL \):
- \( \angle K = 90^\circ \)
- Let's calculate \( \angle M \) and \( \angle L \) again. Wait, if \( \angle K = 90^\circ \), and let's see the angles: if we consider the angles, maybe the correspondence is different. Wait, maybe \( \angle Y = 51^\circ \) and \( \angle L = 48^\circ \) don't match? Wait, no, maybe I made a mistake in angle calculation. Wait, no, let's use the side lengths to check the ratios.

First, check the right - angled triangles. Triangle \( WZY \) is right - angled at \( Z \), with legs \( ZW = 5 \), \( ZY = 4 \), hypotenuse \( WY = 6.4 \). Triangle \( MKL \) is right - angled at \( K \), with legs \( MK = 8.2 \), \( KL = 7.4 \), hypotenuse \( ML = 11 \).

Wait, let's check the ratios of the sides. Let's see the ratios of the corresponding sides. Let's assume that triangle \( WZY \sim \) triangle \( MKL \) (or the other way around). Let's check the angles again. Wait, in triangle \( WZY \), angles are \( 90^\circ \), \( 39^\circ \), \( 51^\circ \). In triangle \( MKL \), angles: \( 90^\circ \), \( 42^\circ \), \( 48^\circ \). These angle measures don't match. Wait, this can't be. Wait, maybe there's a typo in my angle calculation. Wait, no, let's re - examine the triangle \( MKL \). The angle at \( M \) is \( 42^\circ \), angle at \( K \) is \( 90^\circ \), so angle at \( L \) is \( 180 - 90 - 42=48^\circ \). In triangle \( WZY \), angle at \( Y \) is \( 51^\circ \), angle at \( W \) is \( 39^\circ \), angle at \( Z \) is \( 90^\circ \). So the angles are not the same. Wait, but that contradicts. Wait, maybe I misread the angles. Wait, looking at the triangle \( WZY \): \( \angle W = 39^\circ \), \( \angle Z = 90^\circ \), \( \angle Y = 51^\circ \). Triangle \( MKL \): \( \angle M = 42^\circ \), \( \angle K = 90^\circ \), \( \angle L = 48^\circ \). These angle sets are different, so the triangles are not similar? But that seems odd. Wait, maybe I made a mistake. Wait, let's check the side lengths. Let's calculate the ratios of the sides.

For triangle \( WZY \): sides are \( ZW = 5 \), \( ZY = 4 \), \( WY = 6.4 \).

For triangle \( MKL \): sides are \( MK = 8.2 \), \( KL = 7.4 \), \( ML = 11 \).

Let's check the ratios:

\( \frac{ZW}{KL}=\frac{5}{7.4}\approx0.6757 \)

\( \frac{ZY}{MK}=\frac{4}{8.2}\approx0.4878 \)

\( \frac{WY}{ML}=\frac{6.4}{11}\approx0.5818 \)

These ratios are not equal. Also, the angles are not the same (since \( 39^\circ eq42^\circ \), \( 51^\circ eq48^\circ \)). So the triangles are not similar.

Wait, but maybe I mis - identified the corresponding angles. Let's try to find the correct correspondence. Let's see, in triangle \( WZY \), right angle at \( Z \), so the sides: \( ZW \) and \( ZY \) are legs, \( WY \) is hypotenuse. In triangle \( MKL \), right angle at \( K \), so \( MK \) and \( KL \) are legs, \( ML \) is hypotenuse.

Wait, maybe the angle at \( Y = 51^\circ \) and angle at \( M = 42^\circ \)? No, that doesn't match. Wait, perhaps there's an error in the problem or my analysis. Wait, let's re - calculate the angles of triangle \( MKL \) again. If \( \angle K = 90^\circ \), \( \angle L = 48^\circ \), then \( \angle M=180 - 90 - 48 = 42^\circ \) (correct). In triangle \( WZY \), \( \angle Z = 90^\circ \), \( \angle W = 39^\circ \), \( \angle Y = 51^\circ \). So the angle measures are different. Therefore, the triangles do not satisfy the AA (Angle - Angle) similarity criterion (since two angles of one triangle must be equal to two angles of the other triangle for similarity). So the pair of figures is not similar.

Answer:

The pair of triangles is not similar because their corresponding angles are not equal (the angle measures of the two triangles do not match the requirements of the AA similarity criterion, and the ratios of corresponding sides are also not equal).