Question

find the measures of the numbered angles.
(the figure is not to scale.)
the measure of ∠1 is 58 °.
(simplify your answer. type an integer or a decimal.)
the measure of ∠2 is \\(\square\\)°.
(simplify your answer. type an integer or a decimal.)

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). For the triangle with \(\angle1\), \(32^\circ\), and the right angle (but wait, \(\angle2\)'s triangle: angles are \(\angle1 = 58^\circ\), \(54^\circ\), and \(\angle2\). Wait, no—wait, the triangle containing \(\angle2\): angles are \(54^\circ\), \(\angle1 = 58^\circ\)? Wait, no, let's correct. Wait, the triangle with \(\angle2\): we know one angle is \(54^\circ\), another is \(\angle1 = 58^\circ\)? No, wait, the triangle has angles \(54^\circ\), \(\angle1\) (wait, no, \(\angle1\) is in a right triangle? Wait, no, the triangle with \(\angle2\): sum of angles is \(180^\circ\). So \(\angle2 + 54^\circ + \angle1 = 180^\circ\)? Wait, \(\angle1\) is \(58^\circ\), so \(\angle2 = 180 - 54 - 58\)? Wait, no, wait: the triangle with \(\angle2\): angles are \(54^\circ\), \(\angle1 = 58^\circ\)? Wait, no, let's re-express. Wait, the triangle containing \(\angle2\): three angles are \(54^\circ\), \(\angle1 = 58^\circ\), and \(\angle2\). So sum is \(180\). So \(\angle2 = 180 - 54 - 58\)? Wait, \(180 - 54 = 126\), \(126 - 58 = 68\)? No, wait, maybe I messed up. Wait, no—wait, the triangle with \(\angle2\): is it a triangle where one angle is \(54^\circ\), another is \(\angle1 = 58^\circ\)? Wait, no, \(\angle1\) is \(58^\circ\), so \(\angle2 = 180 - 54 - 58 = 68\)? Wait, no, wait, maybe the triangle is: angles are \(54^\circ\), \(\angle1\) (58°), and \(\angle2\). So \(54 + 58 + \angle2 = 180\). So \(112 + \angle2 = 180\), so \(\angle2 = 68\)? Wait, no, wait, maybe I made a mistake. Wait, no—wait, the triangle with \(\angle2\): let's check again. The sum of angles in a triangle is \(180^\circ\). So \(\angle2 + 54^\circ + 58^\circ = 180^\circ\). So \(\angle2 = 180 - 54 - 58 = 68\)? Wait, no, 54 + 58 is 112, 180 - 112 is 68. Wait, but maybe I confused the angles. Wait, no, the problem: \(\angle1\) is \(58^\circ\) (from the right triangle: 90 + 32 + \(\angle1\) = 180, so \(\angle1 = 58\), correct). Then the triangle with \(\angle2\): angles are \(54^\circ\), \(\angle1 = 58^\circ\), and \(\angle2\). So sum is 180. So \(\angle2 = 180 - 54 - 58 = 68\)? Wait, no, 54 + 58 is 112, 180 - 112 is 68. So \(\angle2 = 68^\circ\). Wait, but let's do it step by step.

Step1: Confirm \(\angle1\)

In the right triangle (with 32° and right angle), \(\angle1 + 32^\circ + 90^\circ = 180^\circ\). So \(\angle1 = 180 - 90 - 32 = 58^\circ\) (given, correct).

Step2: Find \(\angle2\)

In the triangle with \(\angle2\), \(54^\circ\), and \(\angle1 = 58^\circ\), sum of angles is \(180^\circ\). So:

$$ \angle2 + 54^\circ + 58^\circ = 180^\circ $$

Subtract \(54^\circ\) and \(58^\circ\) from \(180^\circ\):

$$ \angle2 = 180 - 54 - 58 = 68^\circ $$

Answer:

\(68\)