Question

for each of the following pairs of triangles, (a) identify if they are similar or not similar. (b) if so, give a reason why they are similar (aa, sss, or sas similarity), and (c) complete the similarity statement.
1.
2.
3.
(a) similar or not similar?
(a) similar or not similar?
(a) similar or not similar?
(b) aa, ss, sas, or not similar?
(b) aa, ss, sas, or not similar?
(b) aa, ss, sas, or not similar?
(c) \\( \triangle p q r \sim \triangle \\)______ or
(c) \\( \triangle c d e \sim \triangle \\)______ or
(c) \\( \triangle b c d \sim \triangle \\)______ or
not similar
not similar
not similar

Explanation:

1.

(a)

Step1: Check angle - angle similarity

Since \(\angle QPR=\angle EPF\) (vertically opposite angles) and \(\frac{PQ}{PE}=\frac{21}{7} = 3\), \(\frac{PR}{PF}=\frac{21}{7}=3\). Also, the included angles are equal.

Step2: Apply SAS similarity

If two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, the triangles are similar.

Step1: Check for angle - angle similarity

We know that \(\angle DCE=\angle FCK\) (vertically opposite angles). But we do not have information about the proportionality of sides or another pair of equal angles.

Step1: Check angle - angle similarity

We have \(\angle B=\angle U = 30^{\circ}\). Now, \(\frac{BC}{US}=\frac{11}{5.5}=2\), \(\frac{BD}{UT}=\frac{10}{5} = 2\). The included angles (\(\angle B\) and \(\angle U\)) are equal.

Step2: Apply SAS similarity

Since two sides of \(\triangle BCD\) are proportional to two sides of \(\triangle UTS\) and the included angles are equal.

Answer:

(a) Similar
(b) SAS similarity
(c) \(\triangle PQR\sim\triangle PEF\)

2.

(a)