Question

The second angle of a triangle is 20 more than the first angle. The third angle is two times the first. Find the three angles.

Answer 1

`\large\boxed{\textsf{First Angle = 40}^(\circ)}`

`\large\boxed{\textsf{Second Angle = 60}^(\circ)}`

`\large\boxed{\textsf{Third Angle = 80}^(\circ)}`

Explanation:

`\textsf{We are asked to find the measurement of 3 unknown angles. We are given that}`

`\textsf{a Triangle is the shape, meaning that there are 3 angles total.}`

`\large\underline{\textsf{What is a Triangle?}}`

`\textsf{A Triangle is a 3-sided shape with 3 angles. Sometimes, these can be congruent.}`

`\textsf{Because a Triangle has 3 angles, the sum of the angles' measurements is equal}`

`\textsf{to 180}^(\circ).`

`\large\underline{\textsf{Forming an Equation;}}`

`\textsf{We know that the first angle is not stated to relate to any other angles, hence}`

`\textsf{let's call this angle \boxed{\tt x.}}`

`\textsf{The Second Angle is 20 more than the first angle, or x. This is represented as}`

`\boxed{\tt x + 20.}`

`\textsf{The Third Angle is twice the measurement of the first angle. This is represented}`

`\textsf{as;} \ \boxed{\tt 2x.}`

`\textsf{Remember that these angles add up to 180}^(\circ), \ \textsf{hence their combined sum is}`

`\textsf{identified.}`

`\underline{\textsf{Our Equation;}}`

`\boxed{\tt x^(\circ) + x^(\circ) + 20^(\circ) + 2x^(\circ) = 180^(\circ)}`

`\large\underline{\textsf{Solving;}}`

`\textsf{Let's begin by solving for x. Afterwards we can find the measures of all the angles}`

`\textsf{by Substitution.}`

`\underline{\textsf{Solving for x;}}`

`\tt x^(\circ) + x^(\circ) + 20^(\circ) + 2x^(\circ) = 180^(\circ)`

`\textsf{We first should consider that there are like terms in this equation. All the x's can}`

`\textsf{combine together since they're alike.}`

`\tt \boxed{\tt x^(\circ) + x^(\circ)} + 20^(\circ) + \boxed{\tt 2x^(\circ)} = 180^(\circ)`

`\underline{\textsf{This results as;}}`

`\tt 4x^(\circ) + 20^(\circ)= 180^(\circ)`

`\textsf{Our next step should be isolating x, this involves removing 20 from the left side.}`

`\textsf{This involves using the Properties of Equalities which state that whenever a}`

`\textsf{constant is used to manipulate an equation, the expressions still show equality.}`

`\textsf{For our problem, using the Subtraction Property of Equality, when we subtract}`

`\textsf{20 from both sides of the equation, then the equation remains equal.}`

`\underline{\textsf{Subtract 20 from both sides of the equation;}}`

`\tt 4x^(\circ) + 20^(\circ) - 20^(\circ) = 180^(\circ) - 20^(\circ)`

`\tt 4x^(\circ) = 160^(\circ)`

`\textsf{Using the Division Property of Equality, we are able to divide each side by 4 to}`

`\textsf{remove the coefficient of 4 from x.}`

`\tt (4x^(\circ))/(4) = (160^(\circ))/(4)`

`\large\boxed{\tt x = 40^(\circ)}`

`\large\underline{\textsf{Finding the Unknown Angles;}}`

`\textsf{We know that the measure of x is 40, which represents the measure of the first}`

`\textsf{angle.}`

`\large\boxed{\textsf{First Angle = 40}^(\circ)}`

`\textsf{We know that the second angle is 20 more than the first angle. Knowing that}`

`\textsf{the first angle is 40, the sum is 60.}`

`\large\boxed{\textsf{Second Angle = 60}^(\circ)}`

`\textsf{We know that the third angle is twice the measure of the first angle. This means}`

`\textsf{40 is multiplied by 2, which gives us a product of 80.}`

`\large\boxed{\textsf{Third Angle = 80}^(\circ)}`