Question

work with a partner. find each angle measure in each figure. use equations to justify your answers.
a.
b.
c.
d.

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\).
For part a:
We have the equation \(3q + q+q=180\).
Combining like terms, we get \(5q = 180\).
Dividing both sides by 5, \(q=\frac{180}{5}=36\).
So the angles are \(3q = 3\times36 = 108^{\circ}\), \(q = 36^{\circ}\), \(q = 36^{\circ}\).
For part b:
The equation is \(m+(m + 10)+2m=180\).
Combining like terms: \(4m+10 = 180\).
Subtracting 10 from both sides: \(4m=180 - 10=170\).
Dividing by 4: \(m=\frac{170}{4}=42.5\).
So the angles are \(m = 42.5^{\circ}\), \(m + 10=42.5+10 = 52.5^{\circ}\), \(2m=2\times42.5 = 85^{\circ}\).
For part c:
The equation is \(x+(x + 10)+(x + 5)=180\).
Combining like terms: \(3x+15 = 180\).
Subtracting 15 from both sides: \(3x=180 - 15=165\).
Dividing by 3: \(x=\frac{165}{3}=55\).
So the angles are \(x = 55^{\circ}\), \(x + 10=55+10 = 65^{\circ}\), \(x + 5=55+5 = 60^{\circ}\).
For part d:
Since one angle is \(90^{\circ}\) (right - triangle), the equation is \(y+(y - 30)+90=180\).
Combining like terms: \(2y+60 = 180\).
Subtracting 60 from both sides: \(2y=180 - 60=120\).
Dividing by 2: \(y = 60\).
So the angles are \(y = 60^{\circ}\), \(y - 30=60 - 30=30^{\circ}\), \(90^{\circ}\).

Answer:

a. The angles are \(108^{\circ}\), \(36^{\circ}\), \(36^{\circ}\).
b. The angles are \(42.5^{\circ}\), \(52.5^{\circ}\), \(85^{\circ}\).
c. The angles are \(55^{\circ}\), \(65^{\circ}\), \(60^{\circ}\).
d. The angles are \(60^{\circ}\), \(30^{\circ}\), \(90^{\circ}\).