Question

10.

m∠1 =

m∠4 =

m∠2 =

m∠5 =

m∠3 =

m∠6 =

Explanation:

Step1: Find \( m\angle1 \)

In a triangle, the sum of angles is \( 180^\circ \). The right triangle has a right angle (\( 90^\circ \)) and \( \angle4 = 86^\circ \)? Wait, no, let's look at the angles. Wait, the figure is a rectangle, so all corners are right angles (\( 90^\circ \)). Let's start with \( \angle1 \) and \( \angle4 \). Wait, the triangle with \( \angle1 \), \( \angle2 \), and the right angle? Wait, maybe \( \angle1 + \angle4 + 90^\circ = 180^\circ \)? No, wait, the angle labeled \( 86^\circ \) and \( \angle4 \): wait, \( \angle4 \) and \( 86^\circ \) are supplementary? No, maybe \( \angle4 = 180^\circ - 86^\circ = 94^\circ \)? No, that can't be. Wait, maybe I misread. Let's re-examine.

Wait, the figure is a rectangle, so opposite sides are equal, and all angles are \( 90^\circ \). Let's look at the left triangle: angle \( 52^\circ \), right angle, so the other angle \( \angle6 \) would be \( 90^\circ - 52^\circ = 38^\circ \)? Wait, no, let's do each angle step by step.

For \( m\angle1 \):

In the right triangle (since it's a rectangle, the corner is \( 90^\circ \)), and we have an angle of \( 86^\circ \) adjacent? Wait, maybe \( \angle1 + 86^\circ + 90^\circ = 180^\circ \)? No, that would make \( \angle1 = 4^\circ \)? Wait, no, maybe \( \angle4 = 180^\circ - 86^\circ = 94^\circ \)? No, that's obtuse. Wait, maybe the triangle with angles \( \angle1 \), \( \angle2 \), and \( \angle4 \) is a triangle, so \( \angle1 + \angle2 + \angle4 = 180^\circ \), and \( \angle2 \) is part of the right angle, so \( \angle2 + \angle3 = 90^\circ \), and \( \angle3 \) is equal to \( 52^\circ \) (since the left triangle has \( 52^\circ \) and right angle, so \( \angle3 = 52^\circ \))? Wait, maybe \( \angle3 = 52^\circ \) because the left triangle: \( 52^\circ + 90^\circ + \angle6 = 180^\circ \), so \( \angle6 = 38^\circ \), but maybe \( \angle3 = 52^\circ \) (alternate interior angles or something). Wait, let's try:

1. \( m\angle1 \): Let's assume that the triangle with \( \angle1 \), \( \angle4 \), and the angle \( 86^\circ \) is a straight line? No, maybe \( \angle4 = 180^\circ - 86^\circ = 94^\circ \)? No, that's not right. Wait, maybe \( \angle4 = 86^\circ \) is a typo, but no. Wait, the sum of angles in a triangle is \( 180^\circ \). If one angle is \( 86^\circ \), and it's a right triangle? No, the corner is a right angle, so \( \angle1 + 86^\circ = 90^\circ \), so \( \angle1 = 90^\circ - 86^\circ = 4^\circ \). Ah, that makes sense! Because in the right angle ( \( 90^\circ \) ), the angle \( 86^\circ \) and \( \angle1 \) add up to \( 90^\circ \), so \( \angle1 = 90^\circ - 86^\circ = 4^\circ \).

For \( m\angle4 \):

\( \angle4 \) and \( 86^\circ \) are supplementary (they form a linear pair), so \( \angle4 = 180^\circ - 86^\circ = 94^\circ \)? Wait, no, that can't be. Wait, no, if \( \angle1 = 4^\circ \), and \( \angle2 \) is \( 90^\circ - \angle1 = 86^\circ \)? No, this is confusing. Wait, let's start over.

Step 1: \( m\angle1 \)

In the right angle ( \( 90^\circ \) ), the angle adjacent to \( 86^\circ \) is \( \angle1 \), so \( \angle1 + 86^\circ = 90^\circ \) (since they are complementary in the right angle). Thus, \( m\angle1 = 90^\circ - 86^\circ = 4^\circ \).

Step 2: \( m\angle4 \)

\( \angle4 \) and \( 86^\circ \) are supplementary (they form a straight line), so \( m\angle4 = 180^\circ - 86^\circ = 94^\circ \). Wait, but that's obtuse. Alternatively, if \( \angle4 \) is in a triangle with \( \angle1 \) and \( \angle2 \), then \( \angle1 + \angle2 + \angle4 = 180^\circ \), and \( \angle2 = 90^\circ - \angle3 \), and \( \angle3 = 52^\circ \) (from the left triangle, since the left triangle has a right angle and \( 52^\circ \), so \( \angle3 = 52^\circ \) because of alternate interior angles or congruent triangles). So \( \angle2 = 90^\circ - 52^\circ = 38^\circ \). Then \( \angle1 + \angle2 + \angle4 = 180^\circ \), so \( 4^\circ + 38^\circ + \angle4 = 180^\circ \), so \( \angle4 = 138^\circ \). No, that's not right. I must be misinterpreting the figure.

Wait, maybe the left triangle: angle \( 52^\circ \), right angle, so the third angle (let's say \( \angle6 \)) is \( 90^\circ - 52^\circ = 38^\circ \). Then \( \angle3 = \angle6 = 38^\circ \) (alternate interior angles, since the figure is a rectangle, so the diagonals or lines are parallel). Then \( \angle2 = 90^\circ - \angle3 = 90^\circ - 38^\circ = 52^\circ \). Then in the triangle with \( \angle1 \), \( \angle2 \), and \( \angle4 \), we have \( \angle1 + \angle2 + \angle4 = 180^\circ \), and \( \angle4 \) is adjacent to \( 86^\circ \), so \( \angle4 = 180^\circ - 86^\circ = 94^\circ \). Then \( \angle1 + 52^\circ + 94^\circ = 180^\circ \), so \( \angle1 = 34^\circ \). No, this is inconsistent.

Wait, maybe the correct approach is:

- \( \angle1 \): In the right angle ( \( 90^\circ \) ), the angle given is \( 86^\circ \), so \( \angle1 = 90^\circ - 86^\circ = 4^\circ \).
- \( \angle4 \): \( \angle4 \) is supplementary to \( 86^\circ \), so \( \angle4 = 180^\circ - 86^\circ = 94^\circ \).
- \( \angle3 \): The left triangle has a right angle and \( 52^\circ \), so the third angle ( \( \angle6 \) ) is \( 90^\circ - 52^\circ = 38^\circ \). Since \( \angle3 \) is equal to \( \angle6 \) (alternate interior angles), \( \angle3 = 38^\circ \).
- \( \angle2 \): \( \angle2 = 90^\circ - \angle3 = 90^\circ - 38^\circ = 52^\circ \).
- \( \angle5 \): \( \angle5 \) is equal to \( \angle2 \) (alternate interior angles or congruent triangles), so \( \angle5 = 52^\circ \).
- \( \angle6 \): As calculated, \( \angle6 = 90^\circ - 52^\circ = 38^\circ \).

Let's verify:

- \( \angle1 = 4^\circ \), \( \angle2 = 52^\circ \), \( \angle4 = 94^\circ \): \( 4 + 52 + 94 = 150 \), which is not 180. So that's wrong.

Alternative approach: The figure is a rectangle, so the two triangles (left and right) are congruent? Maybe. So \( \angle3 = 52^\circ \), \( \angle6 = 52^\circ \)? No, the left triangle has \( 52^\circ \), right angle, so \( \angle6 = 38^\circ \), so \( \angle3 = 38^\circ \). Then \( \angle2 = 90^\circ - 38^\circ = 52^\circ \). Then in the right triangle (with \( \angle1 \), \( \angle2 \), and \( \angle4 \)), \( \angle1 + \angle2 + \angle4 = 180^\circ \), and \( \angle4 = 180^\circ - 86^\circ = 94^\circ \), so \( \angle1 = 180 - 52 - 94 = 34^\circ \). But then \( \angle1 + 86^\circ = 34 + 86 = 120 \), not 90. I'm confused.

Wait, maybe the angle labeled \( 86^\circ \) is \( \angle4 \), so \( \angle4 = 86^\circ \), then \( \angle1 + \angle4 + 90^\circ = 180^\circ \), so \( \angle1 = 180 - 90 - 86 = 4^\circ \). That makes sense! So \( \angle1 = 4^\circ \), \( \angle4 = 86^\circ \) (I misread the label). Oh! Maybe the angle labeled \( 86^\circ \) is \( \angle4 \), so \( \angle4 = 86^\circ \). Then:

- \( m\angle1 \): In the right triangle, \( \angle1 + \angle4 + 90^\circ = 180^\circ \), so \( \angle1 = 180 - 90 - 86 = 4^\circ \).
- \( m\angle4 \): \( 86^\circ \) (since it's labeled as \( 86^\circ \)).
- \( m\angle3 \): The left triangle has a right angle ( \( 90^\circ \) ) and \( 52^\circ \), so \( \angle3 = 90^\circ - 52^\circ = 38^\circ \) (because the two angles in the right triangle add up to \( 90^\circ \)).
- \( m\angle2 \): \( \angle2 = 90^\circ - \angle3 = 90 - 38 = 52^\circ \).
- \( m\angle5 \): \( \angle5 \) is equal to \( \angle2 \) (since the figure is a rectangle, alternate interior angles or congruent triangles), so \( \angle5 = 52^\circ \).
- \( m\angle6 \): \( \angle6 = 90^\circ - 52^\circ = 38^\circ \) (same as \( \angle3 \)).

Let's check:

- \( \angle1 = 4^\circ \), \( \angle4 = 86^\circ \), right angle: \( 4 + 86 + 90 = 180 \), correct.
- \( \angle3 = 38^\circ \), \( \angle2 = 52^\circ \): \( 38 + 52 = 90 \), correct (since they form the right angle).
- \( \angle6 = 38^\circ \), \( 52^\circ \), right angle: \( 38 + 52 + 90 = 180 \), correct.
- \( \angle5 = 52^\circ \), which is equal to \( \angle2 \), makes sense in the rectangle.

Yes, this makes sense. So:

- \( m\angle1 = 4^\circ \)
- \( m\angle4 = 86^\circ \) (wait, but the label is \( 86^\circ \) next to \( \angle4 \), so that's correct)
- \( m\angle2 = 52^\circ \)
- \( m\angle5 = 52^\circ \)
- \( m\angle3 = 38^\circ \)
- \( m\angle6 = 38^\circ \)

Step1: Calculate \( m\angle1 \)

In the right triangle, sum of angles is \( 180^\circ \). One angle is \( 90^\circ \) (right angle), another is \( 86^\circ \) ( \( \angle4 \) ). So:
\( m\angle1 + 90^\circ + 86^\circ = 180^\circ \)
\( m\angle1 = 180^\circ - 90^\circ - 86^\circ = 4^\circ \)

Step2: Calculate \( m\angle4 \)

\( \angle4 \) is labeled \( 86^\circ \), so \( m\angle4 = 86^\circ \) (or if it’s supplementary, but here it’s part of the triangle, so it’s \( 86^\circ \)).

Step3: Calculate \( m\angle3 \)

In the left right triangle, one angle is \( 52^\circ \), another is \( 90^\circ \). Sum of angles in a triangle is \( 180^\circ \), so:
\( m\angle3 + 52^\circ + 90^\circ = 180^\circ \)
\( m\angle3 = 180^\circ - 52^\circ - 90^\circ = 38^\circ \)

Step4: Calculate \( m\angle2 \)

\( \angle2 \) and \( \angle3 \) form a right angle ( \( 90^\circ \) ), so:
\( m\angle2 + m\angle3 = 90^\circ \)
\( m\angle2 = 90^\circ - 38^\circ = 52^\circ \)

Step5: Calculate \( m\angle5 \)

\( \angle5 \) is congruent to \( \angle2 \) (alternate interior angles in the rectangle), so \( m\angle5 = 52^\circ \)

Step6: Calculate \( m\angle6 \)

\( \angle6 \) and \( 52^\circ \) form a right angle, so:
\( m\angle6 + 52^\circ = 90^\circ \)
\( m\angle6 = 90^\circ - 52^\circ = 38^\circ \)

Answer:

\( m\angle1 = \boldsymbol{4^\circ} \)
\( m\angle4 = \boldsymbol{86^\circ} \)
\( m\angle2 = \boldsymbol{52^\circ} \)
\( m\angle5 = \boldsymbol{52^\circ} \)
\( m\angle3 = \boldsymbol{38^\circ} \)
\( m\angle6 = \boldsymbol{38^\circ} \)