Question

consider one triangle whose sides measure $\frac{3}{2}$ units, $\frac{5}{2}$ units, and 2 units. consider another triangle whose sides measure 2 units, $\frac{5}{2}$ units, and $\frac{3}{2}$ units. are these triangles congruent, similar, or both? explain your answer. the triangles are both congruent and similar. and since congruent triangles have a ratio of 1 for all corresponding side - pairs, they also meet similarity condition. sample response: since there are three pairs of congruent sides, we know the triangles are congruent by the sss congruence theorem. the corresponding sides of the triangle are also in proportion, so they are also similar by the sss similarity theorem. which did you include in your answer? find corresponding sides. use the sss congruence theorem. corresponding sides are the same length. use the sss similarity theorem. the proportion is 1.

Explanation:

• Find corresponding sides: The sides of the first triangle are \(\frac{3}{2}\), \(\frac{5}{2}\), and \(2\). The sides of the second triangle are \(2\), \(\frac{5}{2}\), and \(\frac{3}{2}\). By comparing, we can identify the corresponding sides.
• Use the SSS congruence theorem: Since \(\frac{3}{2}=\frac{3}{2}\), \(\frac{5}{2}=\frac{5}{2}\), and \(2 = 2\) (all corresponding sides are equal), by the SSS (Side - Side - Side) congruence theorem, the two triangles are congruent.
• Corresponding sides are the same length: As shown above, each pair of corresponding sides has the same numerical value.
• Use the SSS similarity theorem: For similarity, if the ratio of corresponding sides is equal. Here, the ratio of each pair of corresponding sides is \(\frac{\frac{3}{2}}{\frac{3}{2}}=\frac{\frac{5}{2}}{\frac{5}{2}}=\frac{2}{2}=1\). By the SSS similarity theorem (if the ratios of the corresponding sides of two triangles are equal, the triangles are similar), the triangles are similar.
• The proportion is 1: As calculated, the ratio of each pair of corresponding sides (e.g., for the first pair of sides \(\frac{\text{side of first triangle}}{\text{side of second triangle}}=\frac{\frac{3}{2}}{\frac{3}{2}} = 1\)) is \(1\).

Answer:

All of the following should be included:

• Find corresponding sides.
• Use the SSS congruence theorem.
• Corresponding sides are the same length.
• Use the SSS similarity theorem.
• The proportion is 1.