4.
m∠1 =
m∠2 =
m∠3 =
6.
m∠1 =
m∠2 =
m∠3 =

Question

4.
 m∠1 =
 m∠2 =
 m∠3 =
6.
 m∠1 =
 m∠2 =
 m∠3 =

Accepted Answer

### Problem 4:
# Explanation:
## Step1: Find $ m\angle 3 $
In a triangle, the sum of angles is $ 180^\circ $. For the triangle with $ 84^\circ $ and $ 43^\circ $, $ m\angle 3 = 180 - 84 - 43 = 53^\circ $? Wait, no, wait. Wait, the angle $ \angle 3 $ and the angle adjacent? Wait, no, looking at the diagram, the two triangles: the top triangle has angles $ 66^\circ $, $ \angle 2 $, and $ \angle 1 $. The bottom triangle has $ \angle 3 $, $ 43^\circ $, and the angle adjacent to $ \angle 2 $. Wait, actually, $ \angle 3 $ and the angle with $ 84^\circ $ and $ 43^\circ $? Wait, no, let's re-examine.

Wait, the bottom triangle: angles are $ \angle 3 $, $ 43^\circ $, and the angle that is supplementary to $ 84^\circ $? No, wait, the angle $ \angle 3 $ and the triangle with $ 84^\circ $ and $ 43^\circ $: actually, $ \angle 3 $ is in a triangle where the other two angles are $ 84^\circ $ (wait, no, the top triangle has $ 66^\circ $, $ \angle 2 $, $ \angle 1 $. The bottom triangle has $ \angle 3 $, $ 43^\circ $, and the angle equal to $ \angle 2 $ (vertical angles? Wait, no, maybe $ \angle 2 $ and the angle in the bottom triangle are vertical? Wait, no, let's start over.

Wait, the top triangle: angles are $ 66^\circ $, $ \angle 2 $, $ \angle 1 $. The bottom triangle: angles are $ \angle 3 $, $ 43^\circ $, and the angle that is $ 180 - 84 = 96^\circ $? No, wait, the angle labeled $ 84^\circ $ and $ \angle 3 $'s triangle: wait, maybe $ \angle 3 $ is in a triangle with $ 84^\circ $ and $ 43^\circ $? Wait, no, the sum of angles in a triangle is $ 180^\circ $. So for the bottom triangle: $ m\angle 3 + 43^\circ + $ (the angle adjacent to $ 84^\circ $) $ = 180^\circ $. But the angle adjacent to $ 84^\circ $ is supplementary to $ 84^\circ $? No, wait, $ 84^\circ $ and the angle in the bottom triangle are vertical? No, maybe $ \angle 2 $ is equal to the angle in the bottom triangle (vertical angles). Wait, let's correct.

Wait, the top triangle: angles are $ 66^\circ $, $ \angle 2 $, $ \angle 1 $. So $ 66 + \angle 2 + \angle 1 = 180 $.

The bottom triangle: angles are $ \angle 3 $, $ 43^\circ $, and $ \angle 2 $ (since they are vertical angles, so equal). So $ \angle 3 + 43 + \angle 2 = 180 $.

Also, the angle with $ 84^\circ $: wait, the angle labeled $ 84^\circ $ is adjacent to $ \angle 2 $? Wait, no, the diagram shows: top triangle has angle $ 66^\circ $, side with $ \angle 2 $, and $ \angle 1 $. The bottom triangle has $ \angle 3 $, $ 43^\circ $, and a side with $ 84^\circ $ adjacent? Wait, maybe $ \angle 3 $ is calculated as $ 180 - 84 - 43 = 53^\circ $? No, that can't be. Wait, no, the angle $ 84^\circ $ and $ \angle 3 $'s triangle: actually, $ \angle 3 $ is in a triangle where the other two angles are $ 84^\circ $ and $ 43^\circ $? Wait, no, sum of angles in a triangle is $ 180 $, so $ m\angle 3 = 180 - 84 - 43 = 53^\circ $? Wait, but then $ \angle 2 $ would be equal to $ 180 - 66 - \angle 1 $, and $ \angle 3 $ is $ 53^\circ $. Wait, maybe I messed up.

Wait, let's look at the bottom triangle: angles are $ \angle 3 $, $ 43^\circ $, and the angle that is $ 180 - 84 = 96^\circ $? No, $ 84^\circ $ and the angle next to it: if $ 84^\circ $ is one angle, then the adjacent angle is $ 180 - 84 = 96^\circ $, but that's a straight line. Wait, no, the angle $ \angle 3 $ is in a triangle with $ 43^\circ $ and the angle equal to $ \angle 2 $ (vertical angles). Wait, maybe the correct approach is:

For the bottom triangle: $ m\angle 3 + 43^\circ + $ (angle equal to $ \angle 2 $) $ = 180^\circ $.

For the top triangle: $ 66^\circ + \angle 2 + \angle 1 = 180^\circ $.

Also, the angle $ 84^\circ $ and the angle in the bottom triangle: wait, $ 84^\circ $ and $ \angle 3 $'s triangle: maybe $ \angle 3 $ is $ 180 - 84 - 43 = 53^\circ $? No, that gives $ 53 $, but let's check $ \angle 2 $.

Wait, the angle $ \angle 2 $ is in the top triangle: $ 66 + \angle 2 + \angle 1 = 180 $.

The angle adjacent to $ 84^\circ $ is $ 180 - 84 = 96^\circ $, but that's not in the triangle. Wait, maybe I made a mistake. Let's try again.

Wait, the bottom triangle: angles are $ \angle 3 $, $ 43^\circ $, and the angle that is $ 180 - 84 = 96^\circ $? No, that's not possible. Wait, the correct way: the sum of angles in a triangle is $ 180^\circ $. So for the bottom triangle, the three angles are $ \angle 3 $, $ 43^\circ $, and the angle that is equal to $ 180 - 66 - \angle 1 $? No, this is confusing. Wait, maybe the angle $ \angle 3 $ is $ 180 - 84 - 43 = 53^\circ $, $ \angle 2 $ is $ 180 - 66 - \angle 1 $, but $ \angle 1 $ is equal to $ \angle 3 $? No, that doesn't make sense. Wait, maybe the two triangles are isoceles? No, the diagram shows:

Top triangle: angles $ 66^\circ $, $ \angle 2 $, $ \angle 1 $.

Bottom triangle: angles $ \angle 3 $, $ 43^\circ $, and the angle equal to $ \angle 2 $ (vertical angles). So $ \angle 3 + 43 + \angle 2 = 180 $, and $ 66 + \angle 2 + \angle 1 = 180 $. Also, $ \angle 1 $ and $ \angle 3 $ are related? Wait, maybe $ \angle 1 = \angle 3 $? No, that's not necessarily. Wait, maybe the angle $ 84^\circ $ is a typo? No, the user provided the diagram with $ 84^\circ $, $ 66^\circ $, $ 43^\circ $.

Wait, let's start over. Let's assume that $ \angle 3 $ is in a triangle with angles $ 84^\circ $ and $ 43^\circ $. Then $ m\angle 3 = 180 - 84 - 43 = 53^\circ $. Then $ \angle 2 $ is in a triangle with $ 66^\circ $ and $ \angle 1 $. But $ \angle 2 $ is also equal to the angle in the bottom triangle that's adjacent to $ \angle 3 $ and $ 43^\circ $. Wait, no, the angle in the bottom triangle adjacent to $ \angle 3 $ and $ 43^\circ $ is $ 180 - 53 - 43 = 84^\circ $, which is equal to the angle labeled $ 84^\circ $ in the top triangle. Ah! So $ \angle 2 $ is equal to $ 84^\circ $? Wait, no, the top triangle has $ 66^\circ $, $ \angle 2 $, and $ \angle 1 $. So $ 66 + \angle 2 + \angle 1 = 180 $. The bottom triangle has $ \angle 3 $, $ 43^\circ $, and $ 84^\circ $ (since $ 180 - 53 - 43 = 84 $). So $ \angle 2 = 84^\circ $ (vertical angles). Then $ m\angle 2 = 84^\circ $. Then $ m\angle 1 = 180 - 66 - 84 = 30^\circ $. And $ m\angle 3 = 180 - 84 - 43 = 53^\circ $? Wait, no, that can't be. Wait, the bottom triangle: angles are $ \angle 3 $, $ 43^\circ $, and $ 84^\circ $? Then $ 53 + 43 + 84 = 180 $? $ 53 + 43 = 96 $, $ 96 + 84 = 180 $. Yes! So $ m\angle 3 = 53^\circ $, $ m\angle 2 = 84^\circ $ (vertical angles with the $ 84^\circ $ angle), and $ m\angle 1 = 180 - 66 - 84 = 30^\circ $.

## Step1: Calculate $ m\angle 3 $
In the bottom triangle, sum of angles is $ 180^\circ $. Angles are $ \angle 3 $, $ 43^\circ $, and $ 84^\circ $ (since vertical angles with the $ 84^\circ $ angle). So $ m\angle 3 = 180 - 84 - 43 = 53^\circ $.

## Step2: Calculate $ m\angle 2 $
$ \angle 2 $ is vertical to the $ 84^\circ $ angle? Wait, no, the $ 84^\circ $ angle and $ \angle 2 $ are vertical? Yes, because they are opposite angles formed by intersecting lines. So $ m\angle 2 = 84^\circ $.

## Step3: Calculate $ m\angle 1 $
In the top triangle, sum of angles is $ 180^\circ $. Angles are $ 66^\circ $, $ \angle 2 = 84^\circ $, and $ \angle 1 $. So $ m\angle 1 = 180 - 66 - 84 = 30^\circ $.

# Answer:
$ m\angle 1 = 30^\circ $, $ m\angle 2 = 84^\circ $, $ m\angle 3 = 53^\circ $

### Problem 6:
# Explanation:
## Step1: Find $ m\angle 1 $
In the triangle with angles $ 61^\circ $ and $ 68^\circ $, the sum of angles is $ 180^\circ $. So $ m\angle 1 = 180 - 61 - 68 = 51^\circ $.

## Step2: Find $ m\angle 2 $
$ \angle 2 $ is vertical to the angle in the triangle with $ 61^\circ $ and $ 68^\circ $ that is not $ \angle 1 $? Wait, no, the right triangle has a right angle (90°), $ \angle 3 $, and $ \angle 2 $. Also, $ \angle 2 $ is vertical to the angle in the triangle with $ 61^\circ $, $ 68^\circ $, and $ \angle 1 $. Wait, the angle opposite to $ \angle 2 $ in the intersection is equal to $ 180 - 61 - 68 = 51^\circ $? No, wait, the triangle with $ 61^\circ $ and $ 68^\circ $: $ m\angle 1 = 180 - 61 - 68 = 51^\circ $. Then $ \angle 2 $ is equal to the angle in that triangle? Wait, no, the right triangle has a right angle (90°), so $ \angle 2 + \angle 3 = 90^\circ $. Also, $ \angle 2 $ is vertical to the angle in the triangle with $ 61^\circ $, $ 68^\circ $, and $ \angle 1 $ that is $ 180 - 61 - 68 = 51^\circ $? Wait, no, the angle at the intersection: the angle opposite to $ \angle 2 $ is equal to $ 180 - 61 - 68 = 51^\circ $? No, $ 61 + 68 + \angle 1 = 180 $, so $ \angle 1 = 51^\circ $. Then $ \angle 2 $ is equal to $ 180 - 61 - 68 = 51^\circ $? No, wait, the right triangle: $ \angle 2 + \angle 3 + 90 = 180 $, so $ \angle 2 + \angle 3 = 90 $. Also, $ \angle 2 $ is vertical to the angle in the triangle with $ 61^\circ $, $ 68^\circ $, and $ \angle 1 $ that is $ 180 - 61 - 68 = 51^\circ $? Wait, no, the angle at the intersection: the angle between $ 61^\circ $ and $ 68^\circ $ is $ \angle 1 = 51^\circ $, and the angle opposite to $ \angle 2 $ is equal to $ \angle 1 $? No, $ \angle 2 $ is vertical to the angle in the triangle with $ 61^\circ $, $ 68^\circ $, and $ \angle 1 $ that is $ 180 - 61 - 68 = 51^\circ $? Wait, no, $ \angle 2 $ is equal to $ 180 - 61 - 68 = 51^\circ $? Then $ \angle 3 = 90 - 51 = 39^\circ $.

## Step1: Calculate $ m\angle 1 $
In the triangle with angles $ 61^\circ $ and $ 68^\circ $:
$ m\angle 1 = 180^\circ - 61^\circ - 68^\circ = 51^\circ $.

## Step2: Calculate $ m\angle 2 $
$ \angle 2 $ is vertical to the angle in the triangle with $ 61^\circ $, $ 68^\circ $, and $ \angle 1 $ (the angle opposite to $ \angle 2 $ at the intersection). Thus, $ m\angle 2 = 180^\circ - 61^\circ - 68^\circ = 51^\circ $ (or since vertical angles are equal, and the angle in that triangle is $ 51^\circ $, $ \angle 2 = 51^\circ $).

## Step3: Calculate $ m\angle 3 $
The right triangle has a right angle ($ 90^\circ $), so:
$ m\angle 3 = 90^\circ - m\angle 2 = 90^\circ - 51^\circ = 39^\circ $.

# Answer:
$ m\angle 1 = 51^\circ $, $ m\angle 2 = 51^\circ $, $ m\angle 3 = 39^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 4:

Step1: Find \( m\angle 3 \)

Step1: Find \( m\angle 1 \)

Step2: Find \( m\angle 2 \)

Step1: Calculate \( m\angle 1 \)

Step2: Calculate \( m\angle 2 \)

Step3: Calculate \( m\angle 3 \)

Answer:

Problem 6: