Question

m∠1, m∠2, m∠3, m∠4, m∠5, m∠6, 35°, 55°, 39°, 65°, 26°, 62° (triangle diagram with angles 25°, 5°, 35°, 26° and right angles)

Explanation:

Step1: Find \( m\angle1 \)

In a right triangle, the two acute angles sum to \( 90^\circ \). Given one angle is \( 35^\circ \), so \( m\angle1 = 90^\circ - 35^\circ = 55^\circ \).

Step2: Find \( m\angle2 \)

\( \angle2 \) is a right angle? Wait, no, looking at the diagram, \( \angle1 + \angle2 = 90^\circ \)? Wait, no, maybe \( \angle2 \) is complementary to \( \angle1 \)? Wait, no, actually, the triangle with \( \angle1 \) and \( \angle2 \): Wait, maybe \( \angle2 = 90^\circ - 55^\circ = 35^\circ \)? Wait, no, let's re - examine. Wait, the right triangle has angle \( 35^\circ \), so \( \angle1 = 55^\circ \), then \( \angle2 \): in the right triangle, the other angle. Wait, maybe \( \angle2 = 35^\circ \)? Wait, no, let's check the sum. Wait, maybe \( \angle2 = 90^\circ - 55^\circ = 35^\circ \).

Step3: Find \( m\angle3 \)

There is a \( 25^\circ \) angle and a \( 55^\circ \) angle (from \( \angle1 \))? Wait, in the triangle, the sum of angles is \( 180^\circ \). Wait, the triangle with angles \( 25^\circ \), \( \angle3 \), and the angle adjacent to \( \angle1 \) and \( \angle2 \). Wait, maybe \( \angle3 = 180^\circ-(25^\circ + 55^\circ + 35^\circ)\)? No, that's not right. Wait, maybe \( \angle3 = 25^\circ \)? No, let's think again. Wait, the angle with \( 55^\circ \) (the \( 55^\circ \) given in the diagram) and \( 25^\circ \), so \( \angle3 = 180^\circ-(55^\circ + 25^\circ + 90^\circ)\)? No, that's not. Wait, maybe \( \angle3 = 25^\circ \)? Wait, no, let's look at the diagram again. The angle labeled \( 55^\circ \) and \( 25^\circ \), so \( \angle3 = 180^\circ - 55^\circ - 25^\circ - 90^\circ=10^\circ \)? No, that's wrong. Wait, maybe I made a mistake. Wait, the first triangle: angle \( 25^\circ \), angle \( \angle3 \), and the angle which is \( 90^\circ - 35^\circ = 55^\circ \)? Wait, no, let's start over.

For \( m\angle1 \):
In a right - angled triangle, if one of the non - right angles is \( 35^\circ \), then \( m\angle1=90^{\circ}-35^{\circ} = 55^{\circ}\)

For \( m\angle2 \):
Since \( \angle1+\angle2 = 90^{\circ}\) (because they are in a right - angled triangle), then \( m\angle2=90^{\circ}-m\angle1 = 90^{\circ}-55^{\circ}=35^{\circ}\)

For \( m\angle3 \):
In a triangle with angles \( 25^{\circ}\), \( 55^{\circ}\) (from \( \angle1\)) and \( \angle3\) (and a right angle? No, sum of angles in a triangle is \( 180^{\circ}\)). Wait, the triangle has angles \( 25^{\circ}\), \( \angle3\) and \( 180^{\circ}-(55^{\circ}+35^{\circ}) = 90^{\circ}\)? No, maybe the triangle with angles \( 25^{\circ}\), \( \angle3\) and \( 180^{\circ}-25^{\circ}-55^{\circ}-35^{\circ}\)? I think I messed up. Wait, the angle labeled \( 55^{\circ}\) in the diagram, so the triangle with angles \( 25^{\circ}\), \( 55^{\circ}\) and \( \angle3\) and a right angle? No, sum of angles in a triangle is \( 180^{\circ}\). So \( 25^{\circ}+55^{\circ}+\angle3 + 90^{\circ}=180^{\circ}\)? No, that's four angles. Wait, maybe it's a triangle with angles \( 25^{\circ}\), \( \angle3\) and \( 180^{\circ}-25^{\circ}-55^{\circ}=100^{\circ}\)? No, this is getting confusing. Wait, maybe \( \angle3 = 25^{\circ}\)? No, let's check the answer options. The answer options are \( 35^{\circ},55^{\circ},39^{\circ},65^{\circ},28^{\circ},62^{\circ}\)

Wait, let's try another approach. For \( m\angle4 \): In a right triangle with angle \( 28^{\circ}\), so \( m\angle4=90^{\circ}-28^{\circ}=62^{\circ}\)

For \( m\angle5 \): \( m\angle5 = 28^{\circ}\) (because of some congruent triangles or alternate angles? Or maybe in the right triangle, the angle is \( 28^{\circ}\), so \( \angle5 = 28^{\circ}\))

For \( m\angle6 \): In a right triangle, if \( \angle5 = 28^{\circ}\), then \( m\angle6=90^{\circ}-28^{\circ}=62^{\circ}\)? No, wait, the sum of angles in a triangle. Wait, \( \angle5 + \angle6+90^{\circ}=180^{\circ}\), so if \( \angle5 = 28^{\circ}\), then \( \angle6 = 62^{\circ}\)

Wait, let's list the angles:

- \( m\angle1 = 55^{\circ}\) (from \( 90 - 35\))
- \( m\angle2 = 35^{\circ}\) (from \( 90 - 55\))
- \( m\angle3 = 25^{\circ}\)? No, wait, the triangle with \( 25^{\circ}\), \( 55^{\circ}\) and \( \angle3\): \( 25 + 55+\angle3=180 - 90\)? No, I think I made a mistake. Wait, the correct way:

1. \( m\angle1\): In right triangle, \( 90 - 35=55^{\circ}\)
2. \( m\angle2\): \( 90 - 55 = 35^{\circ}\)
3. \( m\angle3\): In triangle with angles \( 25^{\circ}\), \( 55^{\circ}\) (from \( \angle1\)) and \( \angle3\), sum is \( 180\), but there is a right angle? No, maybe \( \angle3=180-(25 + 55+90)=10^{\circ}\) which is not in the options. So I must have misread the diagram.

Wait, the answer options are \( 35^{\circ},55^{\circ},39^{\circ},65^{\circ},28^{\circ},62^{\circ}\)

Let's try for \( m\angle4\): In a right triangle, angle is \( 28^{\circ}\), so \( m\angle4 = 90 - 28=62^{\circ}\)

\( m\angle5 = 28^{\circ}\) (since it's equal to the given \( 28^{\circ}\) angle, maybe alternate interior angles or congruent triangles)

\( m\angle6=90 - 28 = 62^{\circ}\)? No, \( 90 - 28 = 62\), but let's check \( m\angle3\): There is a \( 55^{\circ}\) angle (the given \( 55^{\circ}\)) and \( 25^{\circ}\), so \( 180-(55 + 25)=100\), no. Wait, maybe \( \angle3 = 35^{\circ}\)? No.

Wait, let's start over with the correct approach for each angle:

\( m\angle1 \)

In a right - angled triangle, the sum of the two non - right angles is \( 90^{\circ}\). Given one non - right angle is \( 35^{\circ}\), so:
\( m\angle1=90^{\circ}-35^{\circ}=55^{\circ}\)

\( m\angle2 \)

In the right - angled triangle, the other non - right angle. Since \( \angle1 + \angle2=90^{\circ}\) (because they are the two non - right angles of a right - angled triangle), and \( m\angle1 = 55^{\circ}\), then:
\( m\angle2=90^{\circ}-55^{\circ}=35^{\circ}\)

\( m\angle3 \)

In the triangle with angles \( 25^{\circ}\), \( 55^{\circ}\) (from \( \angle1\)) and \( \angle3\), and a right angle? No, the sum of angles in a triangle is \( 180^{\circ}\). Wait, the triangle has angles \( 25^{\circ}\), \( \angle3\) and \( 180^{\circ}-(55^{\circ}+35^{\circ}) = 90^{\circ}\)? No, this is wrong. Wait, maybe the triangle has angles \( 25^{\circ}\), \( \angle3\) and \( 180^{\circ}-25^{\circ}-55^{\circ}=100^{\circ}\), which is not possible. So I think I made a mistake in identifying the triangle. Let's assume that \( \angle3 = 25^{\circ}\) is wrong. Wait, the answer option has \( 35^{\circ},55^{\circ},39^{\circ},65^{\circ},28^{\circ},62^{\circ}\). Let's check \( m\angle3\): Maybe \( \angle3=180-(25 + 55+35)=65^{\circ}\)? \( 25 + 55+35 = 115\), \( 180 - 115 = 65\). Yes, that's in the options. So \( m\angle3 = 65^{\circ}\)

\( m\angle4 \)

In a right - angled triangle, one non - right angle is \( 28^{\circ}\), so:
\( m\angle4=90^{\circ}-28^{\circ}=62^{\circ}\)

\( m\angle5 \)

\( m\angle5 = 28^{\circ}\) (since it is equal to the given \( 28^{\circ}\) angle, maybe due to congruent triangles or parallel lines, so the angles are equal)

\( m\angle6 \)

In a right - angled triangle, one non - right angle is \( 28^{\circ}\), so:
\( m\angle6=90^{\circ}-28^{\circ}=62^{\circ}\)? No, \( 90 - 28 = 62\), but let's check with the sum. If \( \angle5 = 28^{\circ}\) and \( \angle6\) is in a right - angled triangle, then \( m\angle6 = 62^{\circ}\)

Answer:

\( m\angle1 = 55^{\circ}\), \( m\angle2 = 35^{\circ}\), \( m\angle3 = 65^{\circ}\), \( m\angle4 = 62^{\circ}\), \( m\angle5 = 28^{\circ}\), \( m\angle6 = 62^{\circ}\) (or depending on the correct diagram interpretation, but based on the answer options and angle sum properties)