Question

triangle similarity
for each pair of triangles, determine if the triangles are similar (yes) or not (no). if they are similar, then
provide by which similarity theorem - sss-, sas- or aa-.
4.
5.
yes
no
theorem:
yes
no
theorem:
6.
7.
yes
no
theorem:
yes
no
theorem:
solve for the value of x.

1. δngm ~ δegf
2. δklm~δwvu

Explanation:

Step1: Identify Similar Triangles Ratio

Since \(\triangle NGM \sim \triangle EGF\), corresponding sides are proportional. So, \(\frac{NG}{EG}=\frac{GM}{GF}\). We know \(EG = 63\), \(GM = 10\), \(GF = 35\), and \(NG = x\). Wait, actually, let's check the sides again. Wait, \(NM\) and \(EF\)? Wait, no, the sides: \(NG\) and \(EG\), \(GM\) and \(GF\), \(NM\) and \(EF\). Wait, \(GM = 10\), \(GF = 35\), so the ratio of similarity is \(\frac{GM}{GF}=\frac{10}{35}=\frac{2}{7}\)? Wait, no, wait \(NG\) is part of \(EG\), \(EG = EN + NG\), \(EN = 63 - x\)? Wait, no, the diagram: \(E\) to \(G\) is 63, \(N\) is on \(EG\), \(G\) to \(M\) is 10, \(G\) to \(F\) is 35, \(N\) to \(M\) is 20, \(E\) to \(F\) is 70. Wait, maybe better to use \(\frac{NG}{EG}=\frac{NM}{EF}\). So \(NG = x\), \(EG = 63\), \(NM = 20\), \(EF = 70\). So \(\frac{x}{63}=\frac{20}{70}\).

Step2: Solve for x

Simplify \(\frac{20}{70}=\frac{2}{7}\). So \(\frac{x}{63}=\frac{2}{7}\). Cross - multiply: \(7x = 63\times2\). \(63\times2 = 126\), so \(7x = 126\). Divide both sides by 7: \(x=\frac{126}{7}=18\). Wait, but wait another way: ratio of \(GM\) to \(GF\) is \(10/35 = 2/7\), so \(NG/EG = 2/7\), \(EG = 63\), so \(NG = (2/7)\times63 = 18\). Yes, that matches.

Answer:

For problem 8, \(x = 18\)

Now for problem 9:

Step1: Identify Corresponding Sides

Since \(\triangle KLM \sim \triangle WVU\), corresponding sides are proportional. So, \(\frac{KL}{WV}=\frac{LM}{VU}\). Wait, \(KL = 72\), \(WV = 45\), \(VU = 40\), \(LM = 15x + 4\). Wait, no, let's check the triangles: \(\triangle KLM\) and \(\triangle WVU\). So \(KL\) corresponds to \(WV\), \(LM\) corresponds to \(VU\), \(KM\) corresponds to \(WU\). So \(\frac{KL}{WV}=\frac{LM}{VU}\). So \(\frac{72}{45}=\frac{15x + 4}{40}\).

Step2: Simplify and Solve for x

Simplify \(\frac{72}{45}=\frac{8}{5}\). So \(\frac{8}{5}=\frac{15x + 4}{40}\). Cross - multiply: \(8\times40 = 5\times(15x + 4)\). \(320 = 75x + 20\). Subtract 20 from both sides: \(300 = 75x\). Divide both sides by 75: \(x = 4\).