9. Interior and Exterior Angles of Triangles
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The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.
m∠ 1 = 120 ^(∘) and m∠ 2 = 35^(∘)
We want to find m∠ 1 and m∠ 2 in the following diagram.
We will find these measures one at a time.
In this case, the angle of the measure 138^(∘) is an exterior angle of the larger triangle. Let's mark its remote interior angles on the diagram!
We can see that remote interior angles of the 138^(∘) angle are ∠ 1 and the 18^(∘) angle. The measure of the 138^(∘) angle is equal to the sum of its remote interior angles. 138^(∘) = m∠ 1 + 18^(∘) Using the obtained equation, we can find the measure of ∠ 1. 138^(∘) = m∠ 1 + 18^(∘) [0.4em] ⇕ [0.4em] m∠ 1 = 138^(∘) - 18^(∘) = 120^(∘) Therefore, m∠ 1 = 120^(∘)
Now we will find the measure of ∠ 2 knowing that m ∠ 1 = 120^(∘). Note that the angle of the measure 120^(∘) is an exterior angle of the smaller triangle. Let's start by marking the remote interior angles of the 120^(∘) angle on the diagram!
We can see that remote interior angles of the 120^(∘) angle are the 85^(∘) angle and ∠ 2. The measure of the 120^(∘) angle is equal to the sum of its remote interior angles. 120^(∘) = 85^(∘) + m∠ 2 Using the obtained equation, we can find the measure of ∠ 2. 120^(∘) = 85^(∘) + m∠ 2 [0.4em] ⇕ [0.4em] m∠ 2 = 120^(∘) - 85^(∘) = 35^(∘) Therefore, m∠ 2 = 35^(∘)