Question

similarity - item 35517
complete the statement below about the two figures.
they are
because their corresponding angles are

(then there are two triangles, one labeled with a, b, c with angles at b: 81°, at a: 67°, at c: 32°; the other labeled with x, z, and another vertex, with angles at x: 59°, at z: 55°, and the third angle partially visible.)

Explanation:

Step1: Calculate angles of first triangle

Sum of angles in a triangle is \(180^\circ\). For \(\triangle ABC\), angles are \(81^\circ\), \(67^\circ\), and \(180 - 81 - 67 = 32^\circ\).

Step2: Calculate angles of second triangle

For \(\triangle XYZ\), angles given (from visible) and sum: Let's check, if one angle is \(59^\circ\), another \(55^\circ\), then third is \(180 - 59 - 55 = 66^\circ\)? Wait, no, maybe I misread. Wait, first triangle: \(A = 67^\circ\), \(B = 81^\circ\), \(C = 32^\circ\). Second triangle: Let's recalculate. Wait, maybe the second triangle has angles: let's see, if one is \(59^\circ\), another \(55^\circ\), then \(180 - 59 - 55 = 66^\circ\)? No, that doesn't match. Wait, maybe the second triangle's angles: wait, maybe I made a mistake. Wait, first triangle angles: \(67^\circ\), \(81^\circ\), \(32^\circ\) (since \(67 + 81 + 32 = 180\)). Second triangle: let's check the visible angles. Wait, the problem is about similarity. Wait, maybe the second triangle's angles: let's see, if the angles are not equal, then they are not similar. Wait, no, maybe I misread the angles. Wait, the first triangle: \(\angle A = 67^\circ\), \(\angle B = 81^\circ\), \(\angle C = 32^\circ\). Second triangle: suppose \(\angle X = 59^\circ\), \(\angle Z = 55^\circ\), then \(\angle Y = 180 - 59 - 55 = 66^\circ\). Wait, that doesn't match. Wait, maybe the second triangle's angles are different. Wait, no, maybe the problem is that the corresponding angles are not equal, so they are "not similar" because their corresponding angles are "not congruent". Wait, let's re-express.

Wait, first triangle angles: \(67^\circ\), \(81^\circ\), \(32^\circ\). Second triangle: let's calculate again. If \(\angle X = 59^\circ\), \(\angle Z = 55^\circ\), then \(\angle Y = 180 - 59 - 55 = 66^\circ\). So the angles of first triangle: \(32^\circ\), \(67^\circ\), \(81^\circ\). Second triangle: \(55^\circ\), \(59^\circ\), \(66^\circ\). These are not equal, so the triangles are not similar, and their corresponding angles are not congruent (or not equal).

Step1: Check angle sums

For \(\triangle ABC\): \(67 + 81 + 32 = 180\) (correct). For \(\triangle XYZ\): Let's assume the angles are \(59\), \(55\), and \(66\) (since \(59 + 55 + 66 = 180\)). These angles do not match the angles of \(\triangle ABC\) (32, 67, 81). So corresponding angles are not equal.

Answer:

They are \(\boldsymbol{\text{not similar}}\) because their corresponding angles are \(\boldsymbol{\text{not congruent (or not equal)}}\).