Question

question 7 of 10
is △ham ~ △ste? if so, identify the similarity postulate or
theorem that applies.

a. similar - aa
b. similar - sss
c. similar - sas
d. cannot be determined

Explanation:

Step 1: Check side ratios

First, we find the ratios of the corresponding sides. For the sides given: \( \frac{30}{15} = 2 \) and \( \frac{38}{19} = 2 \). So two pairs of sides are in proportion (ratio 2). But we need to check the included angle. However, the triangles are drawn with the angles at \( A \) and \( T \) (the vertices between the sides) - but wait, actually, in the triangles \( \triangle HAM \) and \( \triangle STE \), the sides \( HA = 30 \), \( AM = 38 \) (wait, no, actually \( HA = 30 \), \( HM \)? Wait, no, the labels: \( \triangle HAM \) has sides \( HA = 30 \), \( AM = 38 \)? Wait, no, the triangle is labeled \( H \), \( A \), \( M \), so \( HA \), \( AM \), and \( HM \)? Wait, no, the other triangle is \( S \), \( T \), \( E \), with \( ST = 15 \), \( TE = 19 \). Wait, actually, the sides are \( HA = 30 \), \( ST = 15 \); \( AM = 38 \), \( TE = 19 \). So the ratio of \( HA/ST = 30/15 = 2 \), and \( AM/TE = 38/19 = 2 \). Now, we need to check if the included angle (the angle between these two sides) is equal. The angle at \( A \) in \( \triangle HAM \) and the angle at \( T \) in \( \triangle STE \) - but looking at the diagram, the triangles are drawn with the same angle (the angle between the two sides with the given lengths) - so the included angle is equal (since they are drawn as having the same angle, probably vertical angles or corresponding angles). Wait, but actually, the SAS similarity postulate states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. Here, we have two sides in proportion (ratio 2) and the included angle (the angle between the two sides) should be equal. Since the triangles are drawn with the angle between the sides (at \( A \) and \( T \)) being the same (as per the diagram's drawing), so the included angle is congruent. Therefore, by SAS similarity, the triangles are similar. Wait, but wait, let's confirm: SAS similarity requires two sides in proportion and the included angle congruent. So we have \( \frac{HA}{ST} = \frac{AM}{TE} = 2 \), and the included angle (angle at \( A \) and angle at \( T \)) is congruent (since the triangles are drawn with that angle, so they are equal). Therefore, by SAS similarity, the triangles are similar. Wait, but wait, maybe I misread the sides. Wait, the sides: in \( \triangle HAM \), the sides are \( HA = 30 \), \( HM \)? No, wait, the labels: \( H \), \( A \), \( M \) - so \( HA \), \( AM \), and \( HM \). In \( \triangle STE \), \( S \), \( T \), \( E \) - so \( ST \), \( TE \), and \( SE \). Wait, but the given lengths are \( HA = 30 \), \( ST = 15 \); \( AM = 38 \), \( TE = 19 \). So the sides \( HA \) and \( ST \), \( AM \) and \( TE \) - the included angle is at \( A \) (between \( HA \) and \( AM \)) and at \( T \) (between \( ST \) and \( TE \)). If those angles are equal (which they appear to be from the diagram, as the triangles are drawn with the same angle), then SAS applies. So the ratio of two sides is equal (2) and included angle is equal, so SAS similarity.

Step 2: Determine the postulate

So since two sides are in proportion (ratio 2) and the included angle is congruent, the triangles are similar by SAS similarity postulate. So the correct option is C.

Answer:

C. Similar - SAS