practice
example 1
find the measure of each angle.
1. find ( mangle abc ) if ( mangle abd = 70^circ ) and ( mangle dbc = 43^circ ).
2. if ( mangle ebc = 55^circ ) and ( mangle ebd = 20^circ ), find ( mangle 2 ).
3. find ( mangle abd ) if ( mangle abc = 110^circ ) and ( mangle 2 = 36^circ ).
example 2
4. flags the alabama state flag is white and has two diagonal red stripes. if the ( mangle 1 = 112^circ ), what is ( mangle 2 )?
5. construction alan has installed a new window above the entrance of an office building. if ( mangle 2 = 44^circ ), what is ( mangle 1 )?
example 3

Question

Accepted Answer

### 1. Find $ m\angle ABC $ if $ m\angle ABD = 70^\circ $ and $ m\angle DBC = 43^\circ $.
# Explanation:
## Step 1: Identify the angle addition postulate
The measure of $ \angle ABC $ is the sum of $ \angle ABD $ and $ \angle DBC $ because $ \angle ABD $ and $ \angle DBC $ are adjacent angles forming $ \angle ABC $. The angle addition postulate states that $ m\angle ABC = m\angle ABD + m\angle DBC $.
## Step 2: Substitute the given values
Substitute $ m\angle ABD = 70^\circ $ and $ m\angle DBC = 43^\circ $ into the formula: $ m\angle ABC = 70^\circ + 43^\circ $.
## Step 3: Calculate the sum
$ 70^\circ + 43^\circ = 113^\circ $.
# Answer: $ 113^\circ $

### 2. If $ m\angle EBC = 55^\circ $ and $ m\angle EBD = 20^\circ $, find $ m\angle 2 $.
# Explanation:
## Step 1: Identify the angle subtraction postulate
The measure of $ \angle 2 $ (which is $ \angle DBC $) is the difference between $ \angle EBC $ and $ \angle EBD $ because $ \angle EBD $ is part of $ \angle EBC $. The angle subtraction postulate states that $ m\angle 2 = m\angle EBC - m\angle EBD $.
## Step 2: Substitute the given values
Substitute $ m\angle EBC = 55^\circ $ and $ m\angle EBD = 20^\circ $ into the formula: $ m\angle 2 = 55^\circ - 20^\circ $.
## Step 3: Calculate the difference
$ 55^\circ - 20^\circ = 35^\circ $.
# Answer: $ 35^\circ $

### 3. Find $ m\angle ABD $ if $ m\angle ABC = 110^\circ $ and $ m\angle 2 = 36^\circ $.
# Explanation:
## Step 1: Identify the angle subtraction postulate
The measure of $ \angle ABD $ is the difference between $ \angle ABC $ and $ \angle 2 $ (which is $ \angle DBC $) because $ \angle ABD $ and $ \angle DBC $ form $ \angle ABC $. The angle subtraction postulate states that $ m\angle ABD = m\angle ABC - m\angle 2 $.
## Step 2: Substitute the given values
Substitute $ m\angle ABC = 110^\circ $ and $ m\angle 2 = 36^\circ $ into the formula: $ m\angle ABD = 110^\circ - 36^\circ $.
## Step 3: Calculate the difference
$ 110^\circ - 36^\circ = 74^\circ $.
# Answer: $ 74^\circ $

### 4. FLAGS The Alabama state flag is white and has two diagonal red stripes. If the $ m\angle 1 = 112^\circ $, what is $ m\angle 2 $?
# Explanation:
## Step 1: Identify the linear pair postulate
$ \angle 1 $ and $ \angle 2 $ form a linear pair, so they are supplementary. The linear pair postulate states that $ m\angle 1 + m\angle 2 = 180^\circ $.
## Step 2: Solve for $ m\angle 2 $
Rearrange the formula to solve for $ m\angle 2 $: $ m\angle 2 = 180^\circ - m\angle 1 $.
## Step 3: Substitute the given value
Substitute $ m\angle 1 = 112^\circ $ into the formula: $ m\angle 2 = 180^\circ - 112^\circ $.
## Step 4: Calculate the difference
$ 180^\circ - 112^\circ = 68^\circ $.
# Answer: $ 68^\circ $

### 5. CONSTRUCTION Alan has installed a new window above the entrance of an office building. If $ m\angle 2 = 44^\circ $, what is $ m\angle 1 $?
# Explanation:
## Step 1: Identify the right angle or complementary angles (assuming the window has a right angle or the angles are complementary)
From the diagram, it appears that $ \angle 1 $ and $ \angle 2 $ are complementary (they form a right angle, $ 90^\circ $). So the sum of $ m\angle 1 $ and $ m\angle 2 $ is $ 90^\circ $. The formula is $ m\angle 1 + m\angle 2 = 90^\circ $.
## Step 2: Solve for $ m\angle 1 $
Rearrange the formula: $ m\angle 1 = 90^\circ - m\angle 2 $.
## Step 3: Substitute the given value
Substitute $ m\angle 2 = 44^\circ $ into the formula: $ m\angle 1 = 90^\circ - 44^\circ $.
## Step 4: Calculate the difference
$ 90^\circ - 44^\circ = 46^\circ $.
# Answer: $ 46^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1. Find \( m\angle ABC \) if \( m\angle ABD = 70^\circ \) and \( m\angle DBC = 43^\circ \).

Step 1: Identify the angle addition postulate

Step 2: Substitute the given values

Step 3: Calculate the sum

Step 1: Identify the angle subtraction postulate

Step 2: Substitute the given values

Step 3: Calculate the difference

Step 1: Identify the angle subtraction postulate

Step 2: Substitute the given values

Step 3: Calculate the difference

Answer:

2. If \( m\angle EBC = 55^\circ \) and \( m\angle EBD = 20^\circ \), find \( m\angle 2 \).