Question

which triangles are congruent to $\triangle a$?
choose all that apply.
triangle $d$ triangle $e$
which triangles are similar to $\triangle a$?
choose all that apply.
triangle $b$ triangle $c$
triangle $d$ triangle $e$

Explanation:

Congruent Triangles

Step1: Recall the definition of congruent triangles

Congruent triangles have the same size and shape. This means their corresponding sides and angles are equal.

Step2: Analyze triangle \(D\)

By counting the grid - squares for side lengths (assuming each grid - square has side length \(1\)). For triangle \(A\), if we assume the vertical side length \(a = 3\), the base (horizontal - like) side length \(b = 2\), and the hypotenuse \(c=\sqrt{3^{2}+2^{2}}=\sqrt{9 + 4}=\sqrt{13}\). For triangle \(D\), the vertical side length is \(3\), the base (horizontal - like) side length is \(2\), and the hypotenuse is \(\sqrt{3^{2}+2^{2}}=\sqrt{13}\). So, triangle \(D\) is congruent to triangle \(A\).

Step3: Analyze triangle \(E\)

For triangle \(E\), if we count the grid - squares, its side lengths are smaller than those of triangle \(A\). So, triangle \(E\) is not congruent to triangle \(A\).

Similar Triangles

Step1: Recall the definition of similar triangles

Similar triangles have the same shape (corresponding angles are equal) and their side lengths are in proportion (scale factor).

Step2: Analyze triangle \(B\)

For triangle \(B\), if we assume the vertical side length \(a_{B}=2\), the base (horizontal - like) side length \(b_{B}=1\). The ratio of side lengths of triangle \(B\) to triangle \(A\) is \(\frac{2}{3}\) (vertical side) and \(\frac{1}{2}\) (base - like side). Since the ratios are not the same, triangle \(B\) is not similar to triangle \(A\).

Step3: Analyze triangle \(C\)

For triangle \(C\), if we assume the vertical side length \(a_{C}=2\), the base (horizontal - like) side length \(b_{C}=2\). The ratio of side lengths of triangle \(C\) to triangle \(A\) (vertical side \(a = 3\), base - like side \(b = 2\)) is \(\frac{2}{3}\) (vertical side) and \(1\) (base - like side). Since the ratios are not the same, triangle \(C\) is not similar to triangle \(A\).

Step4: Analyze triangle \(D\)

Since triangle \(D\) is congruent (\(scale - factor=1\)), and congruent triangles are a special case of similar triangles (\(scale - factor = k = 1\)), triangle \(D\) is similar to triangle \(A\).

Step5: Analyze triangle \(E\)

For triangle \(E\), if we assume the vertical side length \(a_{E}=1\), the base (horizontal - like) side length \(b_{E}=1\). The ratio of side lengths of triangle \(E\) to triangle \(A\) (vertical side \(a = 3\), base - like side \(b = 2\)) is \(\frac{1}{3}\) (vertical side) and \(\frac{1}{2}\) (base - like side). Since the ratios are not the same, triangle \(E\) is not similar to triangle \(A\).

Answer:

• Congruent to \(\triangle A\): Triangle \(D\)
• Similar to \(\triangle A\): Triangle \(D\)