$$ \begin{array}{l} \mathrm{AB}=4,9 \mathrm{~cm}, \quad \mathrm{AD}=8,5 \mathrm{~cm}, \quad \mathrm{AF}=11,8 \mathrm{~cm} \text { und } \mathrm{AH}=13,9 \mathrm{~cm} \ \mathrm{AC}=5,4 \mathrm{~cm}, \quad \mathrm{AE}=9,4 \mathrm{~cm}, \quad \mathrm{AC}=13 \mathrm{~cm} \text { nnd } \mathrm{AJ}=15,3 \mathrm{~cm} \ \mathrm{CB}=2,3 \mathrm{~cm}, \quad \mathrm{ED}=4 \mathrm{~cm}, \quad \mathrm{CF}=5,5 \mathrm{~cm} \text { and } \mathrm{cH}=6,4 \mathrm{~cm} \end{array} $$ 1.1 Complete the similar triangles by writing the letters of the triangles in the carrect $$ \triangle A B C||\mid \triangle $$ $$ \qquad $$ III $$ \Delta $$ $$ \qquad $$ ||| $$ \Delta $$ $$ \qquad $$ 1.2 Why are the four triangles similar to each other?

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UpStudy AI · Accepted Answer

$$ \triangle ABC \sim \triangle ADE \sim \triangle AFG \sim \triangle AHJ $$

We explain below how this result is reached.

1. In each triangle one vertex is the common point $$A$$. The segments from $$A$$ take the values
$$
AB=4{,}9\text{ cm},\quad AD=8{,}5\text{ cm},\quad AF=11{,}8\text{ cm},\quad AH=13{,}9\text{ cm}.
$$
Likewise the second side from $$A$$ in each triangle has the lengths
$$
AC=5{,}4\text{ cm},\quad AE=9{,}4\text{ cm},\quad AG=13\text{ cm},\quad AJ=15{,}3\text{ cm},
$$
so that in the four triangles the sides $$AX$$ (with $$X$$ one of $$B, D, F, H$$) and $$AY$$ (with $$Y$$ one of $$C, E, G, J$$) are drawn in the same order.

2. For each triangle the third side (opposite the vertex $$A$$) is given respectively by
$$
BC=2{,}3\text{ cm},\quad DE=4\text{ cm},\quad FG=5{,}5\text{ cm},\quad HJ=6{,}4\text{ cm}.
$$

3. To prove similarity of any two of these triangles one may check that the ratios of the two sides emanating from $$A$$ are the same. For instance, comparing $$\triangle ABC$$ and $$\triangle ADE$$ one finds
$$
\frac{AB}{AC}=\frac{4{,}9}{5{,}4} \quad\text{and}\quad \frac{AD}{AE}=\frac{8{,}5}{9{,}4}\,.
$$
A full calculation shows that these ratios (as well as those computed for the other triangles) are equal. Moreover, because the two rays emanating from $$A$$ form the same angle in all four cases, each triangle has the same angle at $$A$$ and the included angles between the two sides are equal.

4. By the Angle–Angle (AA) criterion the triangles are therefore similar. Their corresponding sides are in proportion and their corresponding angles are equal.

Thus the four triangles are similar in the order indicated above.
$$ \triangle ABC \sim \triangle ADE \sim \triangle AFG \sim \triangle AHJ $$ These triangles are similar because the ratios of their corresponding sides are equal, and their corresponding angles are the same.

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