Question

Find the unknown angle measure by solving for the given variable

Answer 1

`\text{m}\angle L = 65\°`

`\text{m}\angle M = 65\°`

`\text{m}\angle N = 50\°`

Explanation:

First, we can solve for the variable a by using two pieces of knowledge:

1. the Isosceles Triangle Theorem: if two sides of a triangle are congruent, then their base angles are also congruent
2. the measures of the interior angles of a triangle add to 180°

Using this knowledge, we can deduce that
`\text{m}\angle L = \text{m} \angle M = 2a + 40`, and we can form the following equation to solve for a:

`2(2a + 40) + 4a = 180`

↓ applying the distributive property ...
`a(b + c) = ab + ac`

`4a + 80 + 4a = 180`

↓ combining like terms

`8a + 80 = 180`

↓ subtracting 80 from both sides

`8a = 100`

↓ dividing both sides by 8

`\boxed{a = 12.5}`

Now that we know what a is, we can solve for the measure of each angle in the triangle.

`\text{m}\angle L = 2a + 40`

`\text{m}\angle L = 2(12.5) + 40`

`\text{m}\angle L = 25 + 40`

`\boxed{\text{m}\angle L = 65\°}`

____________________

`\text{m}\angle M = \text{m}\angle L = 65\°`

`\boxed{\text{m}\angle M = 65\°}`

____________________

`\text{m}\angle N = 4a`

`\text{m}\angle N = 4(12.5)`

`\boxed{\text{m}\angle N = 50\°}`