Question

which rule explains why these scalene triangles are similar?
triangle ihj with side lengths 52, 60 and angle 34° at i. triangle vtu with side lengths 75, 65 and angle 34° at t.
options: sss, sas, aa, none of the above; the triangles cannot be proven similar

Explanation:

Step1: Analyze given triangles

Triangle \( IHJ \) has sides \( 52, 60 \) and angle \( 34^\circ \) between them? Wait, no, let's check angles and sides. Triangle \( IHJ \): sides \( IH = 52 \), \( IJ = 60 \), angle at \( I \) is \( 34^\circ \). Triangle \( TVU \) (wait, \( T \) has angle \( 34^\circ \), sides \( VT = 75 \), \( UT = 65 \). Wait, let's check ratios of sides and included angles.

Wait, first, check if the sides around the \( 34^\circ \) angle are proportional. In triangle \( IHJ \), sides adjacent to \( 34^\circ \) (at \( I \)): \( IH = 52 \), \( IJ = 60 \). In triangle \( T \), sides adjacent to \( 34^\circ \) (at \( T \)): \( VT = 75 \), \( UT = 65 \)? Wait no, maybe I mixed up. Wait, triangle \( IHJ \): angle at \( I \) is \( 34^\circ \), sides \( IH = 52 \), \( IJ = 60 \). Triangle \( T \): angle at \( T \) is \( 34^\circ \), sides \( VT = 75 \), \( UT = 65 \)? Wait, no, let's list the sides:

Wait, triangle \( IHJ \): sides \( IH = 52 \), \( IJ = 60 \), angle at \( I \): \( 34^\circ \).

Triangle \( TVU \) (wait, \( V \), \( T \), \( U \)): sides \( VT = 75 \), \( UT = 65 \), angle at \( T \): \( 34^\circ \).

Wait, let's check the ratios of the sides around the \( 34^\circ \) angle.

In triangle \( IHJ \), sides around \( 34^\circ \) (angle at \( I \)): \( IH = 52 \), \( IJ = 60 \).

In triangle \( T \), sides around \( 34^\circ \) (angle at \( T \)): \( UT = 65 \), \( VT = 75 \).

Wait, let's compute the ratios:

\( \frac{IH}{UT} = \frac{52}{65} = \frac{4}{5} \)

\( \frac{IJ}{VT} = \frac{60}{75} = \frac{4}{5} \)

So the two sides around the \( 34^\circ \) angle are proportional (ratio \( \frac{4}{5} \)) and the included angle is equal (\( 34^\circ \)), so by SAS (Side-Angle-Side) similarity criterion, the triangles are similar.

Wait, SAS similarity: if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the triangles are similar.

So here, angle at \( I \) (in \( IHJ \)) and angle at \( T \) (in \( TVU \)) are both \( 34^\circ \), and the sides around these angles are proportional: \( \frac{IH}{UT} = \frac{52}{65} = \frac{4}{5} \), \( \frac{IJ}{VT} = \frac{60}{75} = \frac{4}{5} \). So SAS similarity applies.

Step2: Eliminate other options

• SSS: We don't know all three sides' ratios. We only have two sides. So SSS is out.
• AA: We only have one angle equal (34°), unless we can find another angle, but we don't have info on other angles. So AA is out.
• "None" is wrong because we have SAS similarity.

So the correct rule is SAS.

Answer:

SAS (the option corresponding to SAS, likely one of the radio buttons, but since the options are SSS, SAS, AA, none, the answer is the SAS option, e.g., if the options are labeled as:

A. SSS

B. SAS

C. AA

D. none of the above; the triangles cannot be proven similar

Then the answer is B. SAS)