Question

triangle similarity
for each pair of triangles, determine if the triangles are similar (yes) or not (no). if they are similar, provide by which similarity theorem — sss~, sas~ or aa~.
4.
triangle diagram with labels l, m, k, c, d, lengths 78, 66, 65, 55
yes □
no □
theorem: _________
5.
triangle diagram with labels d, l, c, b, m, lengths 20, 10, 15, 22, 32
yes □
no □
theorem: _________
6.
two triangle diagrams: one with labels s, t, r, lengths 33, 36, 24; another with labels f, e, g, lengths 22, 16, 24
yes □
no □
theorem: _________
7.
two triangle diagrams: one with labels r, t, s (two angles marked); another with labels v, w, u (two angles marked)
yes □
no □
theorem: _________
solve for the value of x.

1. δngm ~ δegf

triangle diagram with labels e, n, g, m, f, lengths 63, x, 20, 10, 70, 35

1. δklm ~ δwvu

two triangle diagrams: one with labels u, v, w, lengths 40, 45; another with labels k, l, m, lengths 72, 15x + 4

Explanation:

Step1: Identify Similar Triangles Proportions

Since \(\triangle NGM \sim \triangle EGF\), corresponding sides are proportional. So, \(\frac{NG}{EG}=\frac{MG}{FG}\). We know \(NG = x\), \(EG = 63\), \(MG = 20\), \(FG = 35\) (wait, no, let's check the diagram again. Wait, \(MG = 20\), \(FG = 35\)? Wait, no, the vertical side: \(MG\) is 20, \(FG\) is 35? Wait, no, the length from \(G\) to \(F\) is 35? Wait, the diagram shows \(G\) to \(M\) is 10? Wait, maybe I misread. Wait, the diagram: \(E\)---\(N\)---\(G\) with \(EN + NG = EG = 63\), \(NG = x\), \(MG = 20\), \(FG = 35\)? Wait, no, the vertical segment: \(G\) to \(M\) is 10? Wait, maybe the correct proportions: \(\frac{NG}{EG}=\frac{MG}{FG}\). Wait, \(EG = 63\), \(NG = x\), \(MG = 20\), \(FG = 35\)? No, that doesn't make sense. Wait, maybe \(MG = 20\), \(FG = 35\)? Wait, no, let's re-express. Wait, \(\triangle NGM \sim \triangle EGF\), so corresponding sides: \(NG\) corresponds to \(EG\), \(MG\) corresponds to \(FG\), \(NM\) corresponds to \(EF\). Wait, the length \(EG = 63\), \(NG = x\), \(MG = 20\), \(FG = 35\)? Wait, no, the vertical side: from \(G\) to \(M\) is 10? Wait, the diagram has \(G\) to \(M\) as 20? Wait, maybe the correct ratio is \(\frac{NG}{EG}=\frac{MG}{FG}\). Wait, \(EG = 63\), \(NG = x\), \(MG = 20\), \(FG = 35\)? No, that's not. Wait, maybe \(MG = 20\), \(FG = 35\) is wrong. Wait, looking at the diagram: \(E\)---\(N\)---\(G\) (length \(EG = 63\)), \(N\)---\(G\) is \(x\), \(G\)---\(M\) is 20, \(M\)---\(F\) is... Wait, the vertical segment: \(G\) to \(F\) is 35? Wait, no, the length from \(G\) to \(F\) is 35? Wait, maybe the correct proportion is \(\frac{NG}{EG}=\frac{MG}{FG}\). So \(\frac{x}{63}=\frac{20}{35}\)? Wait, 20/35 simplifies to 4/7. Then \(x = 63\times\frac{4}{7}= 36\)? Wait, no, 63 divided by 7 is 9, 9 times 4 is 36. Wait, but let's check again. Wait, maybe \(MG = 20\), \(FG = 35\) is incorrect. Wait, the diagram: \(G\) to \(M\) is 20, \(M\) to \(F\) is 15? No, the numbers: \(EG = 63\), \(NG = x\), \(MG = 20\), \(FG = 35\)? Wait, 20/35 = 4/7, 63*(4/7)=36. So \(x = 36\)? Wait, no, maybe I mixed up the sides. Wait, \(\triangle NGM \sim \triangle EGF\), so angle at \(G\) is common? Wait, no, similar triangles, so corresponding angles. So the sides adjacent to the common angle? Wait, maybe the correct proportion is \(\frac{NG}{EG}=\frac{MG}{FG}\). So \(NG = x\), \(EG = 63\), \(MG = 20\), \(FG = 35\). Then \(\frac{x}{63}=\frac{20}{35}\). Simplify 20/35 to 4/7. Then \(x = 63\times\frac{4}{7}= 36\). Wait, but let's check with another ratio. \(NM\) to \(EF\): \(NM\) is... Wait, maybe the horizontal sides: \(EN = 63 - x\), \(NM\) is... No, maybe the vertical sides: \(MG = 20\), \(FG = 35\), so ratio 20/35 = 4/7. Then \(NG/EG = 4/7\), so \(x/63 = 4/7\), so \(x = 36\).

Wait, maybe I made a mistake. Wait, let's do it again. \(\triangle NGM \sim \triangle EGF\), so corresponding sides are proportional. So \(\frac{NG}{EG}=\frac{MG}{FG}\). Given \(EG = 63\), \(NG = x\), \(MG = 20\), \(FG = 35\) (wait, 20 and 35: 20/35 = 4/7. 63*(4/7)=36. So \(x = 36\).

Step2: Solve for x

\(\frac{x}{63}=\frac{20}{35}\)

Simplify \(\frac{20}{35}\) to \(\frac{4}{7}\).

So \(x = 63\times\frac{4}{7}\)

\(63\div7 = 9\), \(9\times4 = 36\).

Answer:

\(x = 36\)