Question

geometry
name braxton fitrant
3.5 exterior angle thereom and triangle sum theorem
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find the measure of each angle indicated.
1)
d
70° ?
c 24° e
g
2)
h e a
82° ?
g 42°
3)
t ?
123° 69°
a u s
4)
t ?
150°
e s u
5)
d ?
49° 143°
c b y
6)
b 34°
150° ?
p a c
solve for x.
7)
c 84°
17x - 1 7x + 5
p d b
8)
j 130°
e 16x + 4 f
14x + 6
g
9)
u 84°
10x - 3 21x + 4
t s g
10)
t 85°
5x - 10 11x + 3
s r j

Explanation:

Step1: Identify the problem type

This is a geometry problem involving the Exterior Angle Theorem and Triangle Sum Theorem. Let's solve problem 1 first. In triangle \( DCE \), we know two interior angles: \( \angle D = 70^\circ \) and \( \angle E = 24^\circ \). The exterior angle at \( C \) (angle \( \angle GCD \)) is equal to the sum of the two non - adjacent interior angles.

Step2: Apply the 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 non - adjacent interior angles. So, for triangle \( DCE \), the measure of the exterior angle at \( C \) (the angle with the question mark) is \( \angle D+\angle E \).
Substitute the given values: \( 70^\circ + 24^\circ=94^\circ \).

Step1 (for problem 2): Analyze triangle \( HEG \)

In triangle \( HEG \), we know \( \angle H = 82^\circ \) and \( \angle G = 42^\circ \). First, find the measure of \( \angle HEG \) using the Triangle Sum Theorem. The Triangle Sum Theorem states that the sum of the interior angles of a triangle is \( 180^\circ \). So, \( \angle HEG=180^\circ-(82^\circ + 42^\circ)=180 - 124=56^\circ \). Then, since \( \angle HEG \) and the angle with measure \( 8 \) (let's assume it's a linear pair) are supplementary (they form a straight line), the measure of the angle with the question mark is \( 180^\circ - 56^\circ = 124^\circ \)? Wait, no, maybe I misread. Wait, the angle at \( E \) adjacent to the exterior angle. Wait, actually, the exterior angle at \( E \) (the angle with the question mark) should be equal to the sum of \( \angle H \) and \( \angle G \) by the Exterior Angle Theorem. So \( \angle H+\angle G=82 + 42 = 124^\circ \).

Step1 (for problem 3): Analyze the exterior angle at \( U \)

The exterior angle at \( U \) is \( 123^\circ \). Let the interior angle at \( U \) be \( \angle U \). Since the exterior angle and the interior angle at \( U \) are supplementary, \( \angle U=180^\circ - 123^\circ = 57^\circ \). Then, using the Triangle Sum Theorem in triangle \( TUS \), where we know \( \angle U = 57^\circ \) and \( \angle S=69^\circ \), the measure of \( \angle T \) (the angle with the question mark) is \( 180^\circ-(57^\circ + 69^\circ)=180 - 126 = 54^\circ \).

Step1 (for problem 4): Analyze the exterior angle at \( S \)

The exterior angle at \( S \) is \( 150^\circ \). The interior angle at \( S \) (adjacent to the exterior angle) is \( 180 - 150=30^\circ \). Since the triangle is a right - triangle (angle \( U \) is a right angle, \( 90^\circ \)), using the Triangle Sum Theorem, the measure of \( \angle T \) is \( 180-(90 + 30)=60^\circ \).

Step1 (for problem 5): Analyze the exterior angle at \( B \)

The exterior angle at \( B \) is \( 143^\circ \), so the interior angle at \( B \) is \( 180 - 143 = 37^\circ \). In triangle \( DBC \), we know \( \angle C = 49^\circ \) and \( \angle B = 37^\circ \). Using the Triangle Sum Theorem, \( \angle D=180-(49 + 37)=180 - 86 = 94^\circ \).

Step1 (for problem 6): Analyze the exterior angle at \( A \)

The exterior angle at \( A \) is \( 150^\circ \), so the interior angle at \( A \) is \( 180 - 150 = 30^\circ \). In triangle \( ABC \), we know \( \angle B = 34^\circ \) and \( \angle A = 30^\circ \). The angle with the question mark (at \( C \)): first, find the third angle? Wait, no, the exterior angle at \( C \) (the angle with the question mark) should be equal to the sum of \( \angle A \) and \( \angle B \) by the Exterior Angle Theorem. So \( 30+34 = 64^\circ \).

Step1 (for problem 7): Apply the Exterior Angle Theorem

The exterior angle at \( D \) ( \( 17x - 1 \)) is equal to the sum of the two non - adjacent interior angles \( \angle C \) ( \( 84^\circ \)) and \( \angle B \) ( \( 7x + 5 \)). So, \( 17x-1=84+(7x + 5) \).
Simplify the right - hand side: \( 17x-1=7x + 89 \).
Subtract \( 7x \) from both sides: \( 17x-7x-1=7x - 7x+89 \), which gives \( 10x-1 = 89 \).
Add \( 1 \) to both sides: \( 10x-1 + 1=89 + 1 \), so \( 10x=90 \).
Divide both sides by \( 10 \): \( x = 9 \).

Step1 (for problem 8): Analyze the exterior angle at \( E \)

The exterior angle at \( E \) is \( 130^\circ \), so the interior angle at \( E \) is \( 180 - 130 = 50^\circ \). Then, using the Triangle Sum Theorem in triangle \( EFG \), \( 50^\circ+(14x + 6)+(16x + 4)=180^\circ \).
Combine like terms: \( 50+14x+6 + 16x + 4=180 \).
\( 20x+60 = 180 \).
Subtract \( 60 \) from both sides: \( 20x=120 \).
Divide by \( 20 \): \( x = 6 \).

Step1 (for problem 9): Analyze the exterior angle at \( S \)

The exterior angle at \( S \) ( \( 21x + 4 \)) is equal to the sum of the two non - adjacent interior angles \( \angle U \) ( \( 84^\circ \)) and \( \angle T \) ( \( 10x-3 \)). So, \( 21x + 4=84+(10x - 3) \).
Simplify the right - hand side: \( 21x+4=10x + 81 \).
Subtract \( 10x \) from both sides: \( 21x-10x + 4=10x-10x + 81 \), \( 11x+4 = 81 \).
Subtract \( 4 \) from both sides: \( 11x=77 \).
Divide by \( 11 \): \( x = 7 \).

Step1 (for problem 10): Analyze the exterior angle at \( R \)

The exterior angle at \( R \) ( \( 11x + 3 \)) is equal to the sum of the two non - adjacent interior angles \( \angle T \) ( \( 85^\circ \)) and \( \angle S \) ( \( 5x-10 \)). So, \( 11x + 3=85+(5x - 10) \).
Simplify the right - hand side: \( 11x+3=5x + 75 \).
Subtract \( 5x \) from both sides: \( 11x-5x+3=5x-5x + 75 \), \( 6x+3 = 75 \).
Subtract \( 3 \) from both sides: \( 6x=72 \).
Divide by \( 6 \): \( x = 12 \).

Answer:

1. \( 94^\circ \)
2. \( 124^\circ \) (using Exterior Angle Theorem: \( 82^\circ+42^\circ = 124^\circ \))
3. \( 54^\circ \)
4. \( 60^\circ \)
5. \( 94^\circ \)
6. \( 64^\circ \)
7. \( x = 9 \)
8. \( x = 6 \)
9. \( x = 7 \)
10. \( x = 12 \)