Question

if these two figures are similar, what is the mea (with two triangles, one with angles 35°, 85°, 60° and the other with a missing angle ?)

Explanation:

Step1: Recall similar figures property

Similar figures have corresponding angles equal. So the triangles have the same angle measures.

Step2: Identify the corresponding angle

The first triangle has angles \(35^\circ\), \(85^\circ\), \(60^\circ\). The second triangle's unknown angle should correspond to one of these. By the order of the angles (the vertex angle of the first triangle, the base angles, and the second triangle's vertex angle), the unknown angle corresponds to \(35^\circ\) (or we can check: sum of angles in a triangle is \(180^\circ\), but since they are similar, corresponding angles are equal. The first triangle's angles are \(35^\circ\), \(85^\circ\), \(60^\circ\). The second triangle's angle at the bottom (the vertex) should match the first triangle's top angle (\(35^\circ\)) or we can see the orientation. Alternatively, since similar triangles have congruent corresponding angles, the angle with "?" should be equal to the \(35^\circ\) angle (or we can check the other angles: if we match the angles, the first triangle's angles are \(35^\circ\) (top), \(85^\circ\) (left base), \(60^\circ\) (right base). The second triangle is rotated, so the bottom angle (vertex) corresponds to the top angle of the first, so \(35^\circ\). Wait, or maybe the \(60^\circ\)? Wait no, let's check the sum. Wait, no, similar triangles have the same angle measures, so the angles are \(35^\circ\), \(85^\circ\), \(60^\circ\) in both. So the unknown angle is equal to the corresponding angle. Looking at the first triangle, the angles are \(35^\circ\) (top), \(85^\circ\) (left), \(60^\circ\) (right). The second triangle is rotated, so the bottom angle (the vertex) is the corresponding angle to the top angle of the first, so \(35^\circ\)? Wait, no, maybe the \(60^\circ\)? Wait, no, let's see the angles. Wait, the first triangle: angles are \(35^\circ\), \(85^\circ\), \(60^\circ\). The second triangle: let's see the sides. Wait, no, the key is similar triangles have equal corresponding angles. So the angle with "?" is equal to the \(35^\circ\) angle (or maybe \(60^\circ\)? Wait, no, let's check the sum. Wait, sum of angles in a triangle is \(180^\circ\). For the first triangle: \(35 + 85 + 60 = 180\), which works. For the second triangle, the unknown angle plus the other two (which correspond to \(85^\circ\) and \(60^\circ\)) should be \(180\). But since they are similar, the angles are the same. So the unknown angle is equal to the \(35^\circ\) angle (the top angle of the first triangle, which corresponds to the bottom angle of the second). So the measure is \(35^\circ\) (or wait, maybe \(60^\circ\)? Wait, no, let's look at the first triangle: the angles are \(35^\circ\) (top), \(85^\circ\) (left), \(60^\circ\) (right). The second triangle is rotated, so the left angle (if we rotate) would be \(85^\circ\), right angle \(60^\circ\), and bottom angle \(35^\circ\). So the unknown angle is \(35^\circ\).

Answer:

\(35^\circ\)