Question

the triangle sum theorem states that the interior angles of any triangle always add to _.
triangle interior angles formula: _ + _ + _ = _
write and solve an equation to find the value of x and/or the missing angle measure in each triangle below.
a)
b)
c)
d)
e)

Explanation:

Step1: Apply the Triangle Sum Theorem for part a)

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is \(180^{\circ}\).
For the triangle in part a), we have the equation \(32^{\circ}+43^{\circ}+x^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(75^{\circ}+x^{\circ}=180^{\circ}\).
Subtract \(75^{\circ}\) from both sides: \(x = 180 - 75=105\).

Step2: Apply the Triangle Sum Theorem for part b)

For the triangle in part b), using the theorem \(x^{\circ}+65^{\circ}+20^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(x + 85=180\).
Subtract \(85\) from both sides: \(x=180 - 85 = 95\).

Step3: Apply the Triangle Sum Theorem for part c)

For the triangle in part c), \(x^{\circ}+59^{\circ}+21^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(x+80 = 180\).
Subtract \(80\) from both sides: \(x=180 - 80=100\).

Step4: Analyze the right - triangle in part d)

In right - triangle \(ABC\) (\(\angle B = 90^{\circ}\)), \(\angle C=31^{\circ}\). Then \(\angle BAC=180^{\circ}-90^{\circ}-31^{\circ}=59^{\circ}\).
Since \(\angle BAC=(3x - 1)^{\circ}\), we set up the equation \(3x-1 = 59\).
Add \(1\) to both sides: \(3x=60\).
Divide both sides by \(3\): \(x = 20\).
\(m\angle A=(3x - 1)^{\circ}=(3\times20 - 1)^{\circ}=59^{\circ}\), \(m\angle B = 90^{\circ}\), \(m\angle C=31^{\circ}\).

Step5: Apply the Triangle Sum Theorem for part e)

For triangle \(DEF\), \(x^{\circ}+3x^{\circ}+2x^{\circ}=180^{\circ}\).
Combine like terms: \(6x=180\).
Divide both sides by \(6\): \(x = 30\).
\(m\angle D=2x^{\circ}=2\times30^{\circ}=60^{\circ}\), \(m\angle E=3x^{\circ}=3\times30^{\circ}=90^{\circ}\), \(m\angle F=x^{\circ}=30^{\circ}\).

Answer:

a) \(x = 105\)
b) \(x = 95\)
c) \(x = 100\)
d) \(x = 20\), \(m\angle A = 59^{\circ}\), \(m\angle B=90^{\circ}\), \(m\angle C = 31^{\circ}\)
e) \(x = 30\), \(m\angle D = 60^{\circ}\), \(m\angle E=90^{\circ}\), \(m\angle F = 30^{\circ}\)