Use the Triangle Angle Sum Theorem to find the largest angle in a triangle with angle measures of (x−20)° , (3x+3)° , and (2x−7)°. Please

Question

GPT-4o mini · Answer

To find the largest angle in the triangle with the measures $(x - 20)^{\circ}$ , $(3 x + 3)^{\circ}$ , and $(2 x - 7)^{\circ}$ , we use the Triangle Angle Sum Theorem, which states that the sum of the angles in a triangle is always 180 degrees. Therefore, we set up the equation:

$(x - 20) + (3 x + 3) + (2 x - 7) = 180$

Now, we can simplify the left side:

$x - 20 + 3 x + 3 + 2 x - 7 = 180$ Combining like terms:

$(1 x + 3 x + 2 x) + (- 20 + 3 - 7) = 180$ $6 x - 24 = 180$

Next, we solve for $x$ :

$6 x = 180 + 24$ $6 x = 204$ $x = 34$

Now that we have the value of $x$ , we can find the measures of each angle:

$(x - 20)^{\circ} = (34 - 20)^{\circ} = 14^{\circ}$
$(3 x + 3)^{\circ} = (3 \cdot 34 + 3)^{\circ} = (102 + 3)^{\circ} = 105^{\circ}$
$(2 x - 7)^{\circ} = (2 \cdot 34 - 7)^{\circ} = (68 - 7)^{\circ} = 61^{\circ}$

Now we have the angle measures:

$14^{\circ}$
$105^{\circ}$
$61^{\circ}$

The largest angle in the triangle is:

$105^{\circ}$