Question

which pairs of figures are congruent? which pairs are similar?
(first figure: two triangles on grid; congruent? ∘ yes ∘ no; similar? ∘ yes ∘ no)
(second figure: two l - shaped figures on grid; congruent? ∘ yes ∘ no; similar? ∘ yes ∘ no)
(third figure: a circle and an ellipse; congruent? ∘ yes ∘ no; similar? ∘ yes ∘ no)
(fourth figure: a square and a rectangle; congruent? ∘ yes ∘ no; similar? ∘ yes ∘ no)

Explanation:

Step1: Analyze Top - Left Figures (Triangles)

• Congruent Check: Congruent figures have the same shape and size. The larger triangle and the smaller triangle have the same shape (both are right - angled triangles with the same angle proportions) but different sizes. So, they are not congruent (select "No" for congruent).
• Similar Check: Similar figures have the same shape (corresponding angles equal) and their sides are in proportion. Since the two triangles have the same angle measures (they are both right - angled and the other angles are equal as the triangles are similar in shape) and the sides are in proportion (the smaller triangle is a scaled - down version of the larger one), they are similar (select "Yes" for similar).

Step2: Analyze Top - Right Figures (L - shaped)

• Congruent Check: The two L - shaped figures have different sizes. So, they are not congruent (select "No" for congruent).
• Similar Check: The shape of the two L - shaped figures is the same (the proportions of the sides and the angles are the same). One is a scaled - up or scaled - down version of the other. So, they are similar (select "Yes" for similar).

Step3: Analyze Bottom - Left Figures (Circle and Ellipse)

• Congruent Check: A circle and an ellipse have different shapes (a circle has all points equidistant from the center, an ellipse has two different radii). So, they are not congruent (select "No" for congruent).
• Similar Check: Since their shapes are different (circle vs ellipse), they are not similar (select "No" for similar).

Step4: Analyze Bottom - Right Figures (Square and Rectangle)

• Congruent Check: A square (with all sides equal) and a rectangle (with length and width different from the square's side) have different sizes and shapes (the square has four equal sides, the rectangle has two pairs of equal sides with different lengths). So, they are not congruent (select "No" for congruent).
• Similar Check: For similarity, the corresponding angles must be equal and sides must be in proportion. A square has angles of \(90^{\circ}\) and a rectangle also has angles of \(90^{\circ}\), but the ratio of length to width for the square is \(1:1\) and for the rectangle is (length : width) \(

eq1:1\) (assuming the rectangle is not a square). Wait, actually, if we consider the square and the rectangle, the square has side length \(s\) (length = width = \(s\)) and the rectangle has length \(l\) and width \(w\) where \(l
eq w\). The ratio of length to width for the square is \(1\), for the rectangle is \(\frac{l}{w}
eq1\). So, the sides are not in proportion. So, they are not similar (select "No" for similar).

Answer:

• Top - Left: Congruent? No; Similar? Yes
• Top - Right: Congruent? No; Similar? Yes
• Bottom - Left: Congruent? No; Similar? No
• Bottom - Right: Congruent? No; Similar? No