read 6.1 angles of a triangle (textbook pgs. 335-338)
exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. ( mangle 1 = mangle a + mangle b )

example 1: find ( mangle 1 ).
triangle ( rst ) with ( angle r = 80^circ ), ( angle s = 60^circ ), exterior angle ( angle 1 ) at ( t ).
( mangle 1 = mangle r + mangle s ) (exterior angle theorem)
( = 80 + 60 ) (substitution)
( = 140 ) (simplify)

example 2: find ( x ).
triangle ( sqr ) with ( angle r = 55^circ ), exterior angle ( angle pqs = 78^circ ), ( angle s = x ).
( mangle pqs = mangle r + mangle s ) (exterior angle theorem)
( 78 = 55 + x ) (substitution)
( 23 = x ) (subtract 55 from each side)

exercises
find the measures of each numbered angle.
1. triangle ( xyz ) with ( angle x = 50^circ ), ( angle y = 65^circ ), exterior angle ( angle 1 ) at ( z ).
2. triangle ( abc ) with ( angle b = 25^circ ), ( angle a = 35^circ ), angles ( angle 1 ), ( angle 2 ) at ( c ).

find each measure.
5. ( mangle abc ): triangle ( abc ) with ( angle a = 95^circ ), exterior angle at ( c ) is ( 145^circ ), ( angle b = 2x^circ ).
6. ( mangle f ): triangle ( efg ) with exterior angle at ( g ) is ( 58^circ ), ( angle e = x ), ( angle f = x ).

Question

read 6.1 angles of a triangle (textbook pgs. 335-338)
exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. ( mangle 1 = mangle a + mangle b )

example 1: find ( mangle 1 ).
triangle ( rst ) with ( angle r = 80^circ ), ( angle s = 60^circ ), exterior angle ( angle 1 ) at ( t ).
( mangle 1 = mangle r + mangle s ) (exterior angle theorem)
( = 80 + 60 ) (substitution)
( = 140 ) (simplify)

example 2: find ( x ).
triangle ( sqr ) with ( angle r = 55^circ ), exterior angle ( angle pqs = 78^circ ), ( angle s = x ).
( mangle pqs = mangle r + mangle s ) (exterior angle theorem)
( 78 = 55 + x ) (substitution)
( 23 = x ) (subtract 55 from each side)

exercises
find the measures of each numbered angle.
1. triangle ( xyz ) with ( angle x = 50^circ ), ( angle y = 65^circ ), exterior angle ( angle 1 ) at ( z ).
2. triangle ( abc ) with ( angle b = 25^circ ), ( angle a = 35^circ ), angles ( angle 1 ), ( angle 2 ) at ( c ).

find each measure.
5. ( mangle abc ): triangle ( abc ) with ( angle a = 95^circ ), exterior angle at ( c ) is ( 145^circ ), ( angle b = 2x^circ ).
6. ( mangle f ): triangle ( efg ) with exterior angle at ( g ) is ( 58^circ ), ( angle e = x ), ( angle f = x ).

Accepted Answer

### Exercise 1: Find $ m\angle 1 $
# Explanation:
## Step 1: Apply Exterior Angle Theorem
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. For $ \triangle XYZ $, $ m\angle 1 = m\angle X + m\angle Y $.
$ m\angle 1 = 50^\circ + 65^\circ $

## Step 2: Simplify
Add the two angles: $ 50 + 65 = 115 $
$ m\angle 1 = 115^\circ $

# Answer:
$ m\angle 1 = 115^\circ $

### Exercise 2: Find $ m\angle 1 $ and $ m\angle 2 $
#### For $ m\angle 1 $:
# Explanation:
## Step 1: Apply Exterior Angle Theorem
For $ \triangle ABC $, $ m\angle 1 = m\angle A + m\angle B $.
$ m\angle 1 = 35^\circ + 25^\circ $

## Step 2: Simplify
Add the two angles: $ 35 + 25 = 60 $
$ m\angle 1 = 60^\circ $

#### For $ m\angle 2 $:
# Explanation:
## Step 1: Linear Pair
$ \angle 1 $ and $ \angle 2 $ form a linear pair, so $ m\angle 1 + m\angle 2 = 180^\circ $.
$ 60^\circ + m\angle 2 = 180^\circ $

## Step 2: Solve for $ m\angle 2 $
Subtract $ 60^\circ $ from both sides: $ m\angle 2 = 180^\circ - 60^\circ = 120^\circ $

# Answer:
$ m\angle 1 = 60^\circ $, $ m\angle 2 = 120^\circ $

### Exercise 5: Find $ m\angle ABC $ (let $ m\angle ABC = 2x^\circ $)
# Explanation:
## Step 1: Linear Pair at $ C $
The exterior angle at $ C $ is $ 145^\circ $, so the adjacent interior angle $ \angle ACB = 180^\circ - 145^\circ = 35^\circ $.

## Step 2: Triangle Angle Sum Theorem
The sum of angles in a triangle is $ 180^\circ $. For $ \triangle ABC $, $ m\angle A + m\angle ABC + m\angle ACB = 180^\circ $.
$ 95^\circ + 2x + 35^\circ = 180^\circ $

## Step 3: Simplify and Solve for $ x $
Combine like terms: $ 130^\circ + 2x = 180^\circ $
Subtract $ 130^\circ $: $ 2x = 180 - 130 = 50 $
Divide by 2: $ x = 25 $

## Step 4: Find $ m\angle ABC $
$ m\angle ABC = 2x = 2(25^\circ) = 50^\circ $

# Answer:
$ m\angle ABC = 50^\circ $

### Exercise 6: Find $ m\angle F $ (let $ m\angle F = x^\circ $, so $ m\angle E = x^\circ $ as the triangle is isosceles)
# Explanation:
## Step 1: Linear Pair at $ G $
The exterior angle at $ G $ is $ 58^\circ $, so the adjacent interior angle $ \angle EGF = 180^\circ - 58^\circ = 122^\circ $.

## Step 2: Triangle Angle Sum Theorem
For $ \triangle EGF $, $ m\angle E + m\angle F + m\angle EGF = 180^\circ $.
$ x + x + 122^\circ = 180^\circ $

## Step 3: Simplify and Solve for $ x $
Combine like terms: $ 2x + 122 = 180 $
Subtract $ 122 $: $ 2x = 180 - 122 = 58 $
Divide by 2: $ x = 29 $
$ m\angle F = 29^\circ $

# Answer:
$ m\angle F = 29^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Exercise 1: Find \( m\angle 1 \)

Step 1: Apply Exterior Angle Theorem

Step 2: Simplify

Step 1: Apply Exterior Angle Theorem

Step 2: Simplify

For \( m\angle 2 \):

Step 1: Linear Pair

Step 2: Solve for \( m\angle 2 \)

Answer:

Exercise 2: Find \( m\angle 1 \) and \( m\angle 2 \)

For \( m\angle 1 \):