find the measure of each numbered angle.
1.
2.
find each measure.
3. ( mangle 1 )
4. ( mangle 2 )
5. ( mangle 3 )

Question

Accepted Answer

### Problem 2:
# Explanation:
## Step1: Recall triangle angle sum
The sum of angles in a triangle is $180^\circ$. Let the unknown angle be $x$.
## Step2: Calculate the unknown angle
Given two angles are $45^\circ$ and $65^\circ$, so $x = 180^\circ - 45^\circ - 65^\circ$
$x = 180 - 110 = 70^\circ$

# Answer:
The measure of the numbered angle (let's assume it's the top angle) is $70^\circ$

### Problem 3: $m\angle1$
# Explanation:
## Step1: Find the third angle in the left triangle
In a triangle, sum of angles is $180^\circ$. Given angles $58^\circ$ and $39^\circ$, the third angle (let's call it $A$) is $180 - 58 - 39 = 83^\circ$
## Step2: Find $m\angle1$
$\angle1$ and angle $A$ are supplementary (they form a linear pair), so $m\angle1 = 180 - 83 = 97^\circ$

# Answer:
$m\angle1 = 97^\circ$

### Problem 4: $m\angle2$
# Explanation:
## Step1: Use vertical angles property
$\angle1$ and $\angle2$ are vertical angles? Wait, no, wait. Wait, actually, $\angle1$ and $\angle2$ are vertical angles? Wait, no, let's re - check. Wait, the angle adjacent to $\angle1$ in the left triangle: Wait, no, actually, $\angle1$ and $\angle2$ are vertical angles? Wait, no, $\angle1$ and $\angle2$ are vertical angles? Wait, no, let's see. Wait, the two triangles are connected by intersecting lines. So $\angle1$ and $\angle2$ are vertical angles? Wait, no, vertical angles are equal. Wait, no, actually, $\angle1$ and the angle opposite to it (which is $\angle2$)? Wait, no, let's use the triangle angle sum. Wait, no, $\angle1$ and $\angle2$ are vertical angles, so they are equal? Wait, no, wait, in the left triangle, we found the angle adjacent to $\angle1$ is $83^\circ$, so $\angle1 = 97^\circ$, and $\angle2$ is equal to $\angle1$ because they are vertical angles? Wait, no, wait, no. Wait, no, $\angle1$ and $\angle2$ are vertical angles? Wait, the two lines intersect, so $\angle1$ and $\angle2$ are vertical angles, so they are equal. Wait, but that can't be. Wait, no, I made a mistake earlier. Wait, let's re - do.

Wait, in the left triangle, angles are $58^\circ$, $39^\circ$, so the third angle is $180-(58 + 39)=83^\circ$. Then, $\angle1$ and this $83^\circ$ angle form a linear pair, so $\angle1 = 180 - 83=97^\circ$. Now, $\angle1$ and $\angle2$ are vertical angles? Wait, no, $\angle1$ and $\angle2$ are vertical angles, so $m\angle2=m\angle1 = 97^\circ$? Wait, no, that can't be. Wait, no, I think I messed up. Wait, the angle in the right triangle: let's look at the right triangle. The right triangle has an angle of $35^\circ$ and we need to find $\angle3$ later. Wait, no, $\angle1$ and $\angle2$ are vertical angles, so they are equal. So $m\angle2 = 97^\circ$? Wait, no, that seems wrong. Wait, no, let's use the linear pair. Wait, $\angle1$ and the angle next to it (in the left triangle) are supplementary. Then, $\angle2$ is equal to $\angle1$ because they are vertical angles. So $m\angle2=97^\circ$

# Answer:
$m\angle2 = 97^\circ$ (assuming $\angle1$ and $\angle2$ are vertical angles)

### Problem 5: $m\angle3$
# Explanation:
## Step1: Use the triangle angle sum in the right triangle
We know $m\angle2 = 97^\circ$ (from problem 4) and one angle is $35^\circ$. Let $m\angle3=x$. Then $x+35^\circ + m\angle2=180^\circ$ (wait, no, $\angle2$ is not in the right triangle. Wait, the right triangle has angles $35^\circ$, $\angle3$, and the angle opposite to $\angle2$. Wait, $\angle2$ and the angle in the right triangle (let's call it $B$) are supplementary? No, wait, the right triangle has angles: one angle is $35^\circ$, another angle is equal to the angle we found in the left triangle (the $83^\circ$ angle) because they are vertical angles? Wait, no, let's start over.

In the left triangle, angles are $58^\circ$ and $39^\circ$, so the third angle (let's call it $C$) is $180-(58 + 39)=83^\circ$. This angle $C$ and the angle in the right triangle (let's call it $D$) are vertical angles, so $D = 83^\circ$. Now, in the right triangle, we have angles $D = 83^\circ$, $35^\circ$, and $\angle3$. So $m\angle3=180-(83 + 35)=180 - 118 = 62^\circ$

# Explanation (step - by - step):
## Step1: Calculate angle $C$ in the left triangle
Sum of angles in a triangle is $180^\circ$. For the left triangle with angles $58^\circ$ and $39^\circ$, $C=180 - 58 - 39=83^\circ$
## Step2: Use vertical angles to find angle $D$ in the right triangle
Angle $C$ and angle $D$ are vertical angles, so $D = C = 83^\circ$
## Step3: Calculate $m\angle3$
In the right triangle, sum of angles is $180^\circ$. Given angles $D = 83^\circ$ and $35^\circ$, $m\angle3=180-(83 + 35)=62^\circ$

# Answer:
$m\angle3 = 62^\circ$

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QUESTION IMAGE

Question

Explanation:

Problem 2:

Step1: Recall triangle angle sum

Step2: Calculate the unknown angle

Step1: Find the third angle in the left triangle

Step2: Find \(m\angle1\)

Step1: Use vertical angles property

Answer:

Problem 3: \(m\angle1\)