Question

which pairs of figures are congruent? which pairs are similar? congruent? yes no similar? yes no congruent? yes n similar? yes no congruent? yes no similar? yes no congruent? yes no similar? yes no

Explanation:

First pair:

• Congruent?
• Congruent figures have the same shape and size. The two rectangles have different side - length ratios (one is more "squat" and the other is more "tall" in proportion). So, they are not congruent.
• Similar?
• Similar figures have the same shape. For rectangles, similarity requires that the ratios of corresponding sides are equal. Let the sides of the first rectangle be \(l_1,w_1\) and the second be \(l_2,w_2\). If \(\frac{l_1}{w_1}

eq\frac{l_2}{w_2}\), they are not similar. Since their side - length ratios are different, they are not similar.

Second pair:

• Congruent?
• The two circles have the same size (radius), so they are congruent.
• Similar?
• All circles are similar because the ratio of their radii (a characteristic dimension for circles) can be related by a scale factor. Since they are congruent (scale factor \(k = 1\)), they are also similar.

Third pair:

• Congruent?
• The two L - shaped figures have different sizes. So, they are not congruent.
• Similar?
• We can consider them as composed of rectangles. If we assume the smaller L - shaped figure is a scaled - down version of the larger one, and if the ratios of all corresponding linear dimensions (horizontal and vertical segments) are equal, then they are similar. Let the lengths of the segments of the larger L - shape be \(a,b,c\) and the smaller be \(ka,kb,kc\) (\(k<1\)). Since the angles (all right - angles) are the same and the side - length ratios are equal, they are similar.

Fourth pair:

• Congruent?
• The two triangles have different side lengths. So, they are not congruent.
• Similar?
• One is a right - triangle and the other is an isosceles (non - right) triangle. Since their angle measures are different (a right - triangle has a \(90^{\circ}\) angle and the other does not), they are not similar.

Answer:

• First pair: Congruent? No; Similar? No
• Second pair: Congruent? Yes; Similar? Yes
• Third pair: Congruent? No; Similar? Yes
• Fourth pair: Congruent? No; Similar? No