Question

2 - 114. which pairs of triangles below are congruent and/or similar? for each part, explain how you know using an appropriate triangle congruence or similarity condition. note: the diagrams are not necessarily drawn to scale. homework help
2 - 117. solve for the missing lengths in the sets of similar figures below. homework help
a. abcd ~ jklm

Explanation:

Step1: Recall similarity conditions

For two triangles to be similar, their corresponding angles must be equal and corresponding - sides must be in proportion. For similar rectangles, the ratios of corresponding sides are equal.

Step2: Analyze part 2 - 114

For each pair of triangles in 2 - 114:

• Check if the ratios of corresponding sides are the same. For example, if we have two triangles with side lengths \(a_1,b_1,c_1\) and \(a_2,b_2,c_2\), we need to check if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\). Also, we can use angle - angle (AA), side - side - side (SSS), or side - angle - side (SAS) similarity criteria.
• If two pairs of corresponding angles are equal (AA criterion), the triangles are similar. If the ratios of all three pairs of corresponding sides are equal (SSS criterion), the triangles are similar. If the ratio of two pairs of corresponding sides are equal and the included angles are equal (SAS criterion), the triangles are similar.

Step3: Analyze part 2 - 117

Since \(ABCD\sim JKLM\), the ratios of corresponding sides are equal. Let the length of \(AB = 12\) mm, \(AD = 6\) mm, \(JK=x\), and \(JM = 10\) mm.
The ratio of corresponding sides gives us \(\frac{AB}{JK}=\frac{AD}{JM}\). Substituting the values, we have \(\frac{12}{x}=\frac{6}{10}\).
Cross - multiply: \(6x=12\times10\).
Solve for \(x\): \(x = \frac{12\times10}{6}=20\) mm.

Answer:

In 2 - 114, use AA, SSS, or SAS similarity criteria to determine if pairs of triangles are similar. In 2 - 117, \(x = 20\) mm.