Question

in the figure, \\( \triangle a b c \\) is similar to \\( \triangle d e f \\).
what is \\( m \angle 1 ? \\)\\( 167 ^ { \circ } \\)( 139 ^ { \circ } \\)( 118 ^ { \circ } \\)( 103 ^ { \circ } \\)

Explanation:

Step1: Find angle C in △ABC

The sum of angles in a triangle is \(180^\circ\). So, \(m\angle C = 180^\circ - 62^\circ - 77^\circ\)
\(m\angle C = 180 - 139 = 41^\circ\)? Wait, no, wait: \(62 + 77 = 139\), so \(180 - 139 = 41^\circ\)? Wait, no, maybe I made a mistake. Wait, no, the problem is about similar triangles. So △ABC ~ △DEF, so corresponding angles are equal. Then, angle at F in △DEF corresponds to angle C? Wait, no, maybe angle 1 is supplementary to the corresponding angle. Wait, let's re-examine.

First, in △ABC, angles are 62° (A), 77° (B), so angle C is \(180 - 62 - 77 = 41^\circ\). Now, △ABC ~ △DEF, so corresponding angles: let's say angle D corresponds to A (62°), angle E corresponds to B (77°), angle F corresponds to C (41°). Then, angle 1 is adjacent to angle F, forming a linear pair (supplementary). So \(m\angle 1 = 180^\circ - 41^\circ = 139^\circ\)? Wait, no, wait the options include 139°, 103°, etc. Wait, maybe I mixed up the correspondence. Wait, maybe angle F corresponds to angle B? No, similar triangles have corresponding angles equal. Let's check again.

Wait, maybe the correspondence is △ABC ~ △DEF, so angle A corresponds to angle D, angle B to angle E, angle C to angle F. So angle F is equal to angle C. Then, angle 1 is supplementary to angle F, so \(m\angle 1 = 180 - m\angle F\). But first, find angle C: \(180 - 62 - 77 = 41\), so angle F is 41, so angle 1 is 180 - 41 = 139? But the options have 139 as one of them. Wait, but maybe I messed up the angle correspondence. Wait, maybe angle F corresponds to angle B? No, that doesn't make sense. Wait, maybe the triangle DEF has angle D equal to angle A (62°), angle E equal to angle C (41°), and angle F equal to angle B (77°)? No, that would change. Wait, no, similar triangles: the order of the letters matters. So △ABC ~ △DEF means A→D, B→E, C→F. So angle A = angle D, angle B = angle E, angle C = angle F. So angle F = angle C = 41°, then angle 1 is supplementary to angle F, so 180 - 41 = 139°. So the answer should be 139°.

Wait, but let's check again. Sum of angles in △ABC: 62 + 77 + C = 180 → C = 41. Then, since △ABC ~ △DEF, angle F = angle C = 41. Then, angle 1 and angle F are supplementary (linear pair), so angle 1 = 180 - 41 = 139. So the answer is 139°.

Step2: Determine m∠1 using supplementary angles

Since ∠1 and ∠F are supplementary (form a linear pair), \(m\angle 1 = 180^\circ - m\angle F\). We found \(m\angle F = 41^\circ\) (from similar triangles, ∠F = ∠C), so \(m\angle 1 = 180 - 41 = 139^\circ\).

Answer:

139°