Question

Find the unknown angle measure by solving for the given variable

Answer 1

Answer:

$\text{m}\angle L = 65\°$

$\text{m}\angle M = 65\°$

$\text{m}\angle N = 50\°$

Step-by-step 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.

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

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

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

$\boxed{\text{m}\angle L = 65\°}$

____________________

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

$\boxed{\text{m}\angle M = 65\°}$

____________________

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

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

$\boxed{\text{m}\angle N = 50\°}$