Question

can i use equivalent ratios?
and congruence
triangles below is similar. use a flowchart to organize your facts and conclusion for each pair of

Explanation:

Step1: Analyze Triangle \( \triangle IHG \) and \( \triangle JKL \)

For \( \triangle IHG \) (right - angled at \( H \)):

• \( IH = 16 \), \( HG = 12 \), \( IG = 20 \) (by Pythagorean theorem, \( 16^{2}+12^{2}=256 + 144=400=20^{2}\))

For \( \triangle JKL \) (right - angled at \( L \)):

• \( JL = 12 \), \( LK = 9 \), \( JK=\sqrt{12^{2}+9^{2}}=\sqrt{144 + 81}=\sqrt{225}=15\)

Step2: Check Ratios of Corresponding Sides

• Ratio of \( IH\) to \( LK\): \( \frac{IH}{LK}=\frac{16}{9}\)
• Ratio of \( HG\) to \( JL\): \( \frac{HG}{JL}=\frac{12}{12} = 1\)
• Ratio of \( IG\) to \( JK\): \( \frac{IG}{JK}=\frac{20}{15}=\frac{4}{3}\)

Wait, maybe we made a mistake in corresponding sides. Let's re - assign. Let's check the ratios of the legs.
For \( \triangle IHG \), legs are \( IH = 16\) and \( HG = 12\), ratio of legs \( \frac{IH}{HG}=\frac{16}{12}=\frac{4}{3}\)
For \( \triangle JKL \), legs are \( LK = 9\) and \( JL = 12\), ratio of legs \( \frac{LK}{JL}=\frac{9}{12}=\frac{3}{4}\)
Wait, maybe the triangles are similar by SSS similarity if we take the correct corresponding sides. Wait, \( \triangle IHG\): sides \( 12,16,20\) (divide by 4: \( 3,4,5\))
\( \triangle JKL\): sides \( 9,12,15\) (divide by 3: \( 3,4,5\))
Ah, so \( \frac{12}{9}=\frac{16}{12}=\frac{20}{15}=\frac{4}{3}\) (wait, \( \frac{12}{9}=\frac{4}{3}\), \( \frac{16}{12}=\frac{4}{3}\), \( \frac{20}{15}=\frac{4}{3}\))
So the ratios of corresponding sides are equal. So \( \triangle IHG\sim\triangle JKL\) by SSS similarity.

For the second pair (part d):

Step1: Analyze the Triangles

We have two triangles with a common angle at \( V\) (since \( \angle V\) is common and \( \angle VSN=\angle VTM\) (corresponding angles as \( SN\parallel TM\) if we assume the lines are parallel, or by AA similarity if the angles are equal).
Let's check the ratios of sides. The side \( VS = 2.5\), \( VM=2.5 + 9=11.5\)? Wait, no, the given sides: \( SN = 5\), \( TM\) (let's see), \( VN\) and \( VM\): \( VM = 11\)? Wait, the diagram shows \( VS = 2.5\), \( SM = 9\), \( SN = 5\), and \( TM\) (the side of the larger triangle) and \( VN\) (the base of the smaller trapezoid - like part). Wait, if we consider \( \triangle VSN\) and \( \triangle VTM\):
\( \angle V\) is common.
\( \frac{VS}{VM}=\frac{2.5}{2.5 + 9}=\frac{2.5}{11.5}=\frac{5}{23}\)
\( \frac{SN}{TM}=\frac{5}{TM}\) (we need to find \( TM\))
Wait, maybe the problem is to check similarity. For the first pair (part b), the triangles are similar because their side lengths are in proportion (\(12:9:15\) and \(12:16:20\) when simplified to \(3:4:5\) and \(3:4:5\) respectively).

Answer:

For part (b), \( \triangle IHG\) and \( \triangle JKL\) are similar because the ratios of their corresponding sides are equal (\( \frac{12}{9}=\frac{16}{12}=\frac{20}{15}=\frac{4}{3}\)), so they satisfy the SSS (Side - Side - Side) similarity criterion.