Question

directions: use this information to answer parts a and b. the graph shows right triangles abc and ade. the hypotenuse of each triangle lies on the same line. part a explain how you know that the triangles are similar. show hints all right triangles are similar. \\( \overline { a d } \\) and \\( \overline { a b } \\) lie on the same line, as do \\( \overline { a e } \\) and \\( \overline { a c } \\), so they are similar by sss. \\( \triangle a b c \\) is transformed into \\( \triangle a c d \\) using a dilation of \\( \frac { 1 } { 2 } \\), so they are similar. angle a is the same for both triangles and m?c is equal to m?e, so the triangles are similar by aa similarity.

Explanation:

Step1: Analyze the properties of similar triangles

Similar triangles can be proved by AA (Angle - Angle) similarity criterion. In right - angled triangles \(ABC\) and \(ADE\), \(\angle A\) is common to both triangles. Also, since \(\angle C=\angle E = 90^{\circ}\) (given that they are right - angled triangles).

Step2: Check the similarity criteria

For AA similarity, if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. Here, \(\angle A=\angle A\) (common angle) and \(\angle C=\angle E\) (right angles).

Answer:

Angle \(A\) is the same for both triangles and \(m\angle C\) is equal to \(m\angle E\), so the triangles are similar by \(AA\) similarity.