Answers:
1.
Part A. Option a) vertical angles theorem and triangle angle-sum theorem
Part B. Option b) x=80, y=80
2. Option d) 123
3. Options 3 and 4:
Triangle Angle-Sum Theorem
Triangle Exterior Angle Theorem
4. Option c) x=37
Solution:
1. Part A
First, you can find x using the triangle angle-sum theorem: The sum of the interior angles of any triangle must be equal to 180°.
Second, you can apply the vertical angles theorem to find y: the angles opposite by the vertex must be congruent.
Then, the answer is option a) vertical angles theorem and triangle angle-sum theorem.
1. Part B.
First: Triangle angle-sum theorem
The sum of the interior angles of any triangle must be equal to 180°. The interior angles of the triangle in the figure are x°, 30°, and 70°, then:
x°+30°+70°=180°
(x+30+70)°=180°
(x+100)°=180°
x+100=180
Solving for x: Subtracting 100 both sides of the equation:
x+100-100=180-100
x=80
Second: Vertical angles theorem: The angles opposite by the vertex must be congruent:
In the figure, the angles x° and y° are opposite by the vertex, then they must be congruent:
y°=x°
y=x
and x=80, then:
y=80
Answer: Option b) x=80, y=80
2. The <1 is an exterior angle of the triangle in the figure, and according with the Triangle Exterior Angle Theorem, an exterior angle of a triangle must be equal to the sum of the interior angles no adjacents to it:
<1=60°+63°
<1=123°
Answer: Option d) 123
3.
You can apply the Triangle Angle-Sum Theorem, to find the third interior angle of the triangle. With this angle you can find the exterior <1.
You can apply the Triangle Exterior Angle Theorem to find the exterior <1.
Answer: Options 3 and 4:
Triangle Angle-Sum Theorem
Triangle Exterior Angle Theorem
4. Using the Triangle angle-sum theorem:
The sum of the interior angles of any triangle must be equal to 180°. The interior angles of the triangle in the figure are (2x-9)°, x°, and (2x+4)°, then:
(2x-9)°+x°+(2x+4)°=180°
(2x-9+x+2x+4)°=180°
Adding similar terms:
(5x-5)°=180°
5x-5=180
Solving for x: Adding 5 both sides of the equation:
5x-5+5=180+5
Adding:
5x=185
Dividing both sides of the equation by 5:
(5x)/5=185/5
x=37
Answer: Option c) x=37
