Properties of Triangles Examples

Find the remaining interior angle in the triangle.

Since we know that all the interior angles add up to 180° according to the Angle Sum Theorem, we can write an equation and then fill in what we know.

m∠1 + m∠2 + m∠3 = 180
52 + 63 + m∠3 = 180
m∠3 = 65

That means our remaining interior angle is 65°.

Find the measure of ∠1.

While we might look at ∆ABC initially, we only know one of the three angles so we can't use it alone to find ∠1. We could use the Angle Sum Theorem to find ∠BCD and then knowing that ∠1 is supplementary to that, but there's an even easier way.

Since ∠1 is the exterior angle to ∆BCD, and we already know the remote interior angles of that triangle, we can use the Exterior Angle Theorem. That will give us this equation.

m∠1 = m∠CBD + m∠CDB
m∠1 = 17 + 21
m∠1 = 38

That's all we need.

Find the measure of ∠2. 

When we first look at ∆CDE, we only have ∠2 and 31° labeled. However, ∠DCE and ∠1 are vertical angles, and you know what that means: they're as congruent as they'll ever be.

Since we already know that m∠1 = 38, we know that m∠DCE = 38 also. Now that we know two of the angles, we can find m∠2 using the Angle Sum Theorem.

m∠CDE + m∠2 + m∠DCE = 180
31 + m∠2 + 38 = 180
m∠2 = 111

That means ∠2 is a 111° angle.

Find the measure of ∠3.

We already know ∠1, but would it be possible to solve the problem if we didn't? Sure thang!

Take a look at ∆ABD. One angle is 21°, another is ∠3, and the top angle is 85° + 17°. (We can do that because of the Angle Addition Postulate.) So, in case we've forgotten that ∠1 is 38°, we can still solve this using the Angle Sum Theorem.

m∠3 + m∠BDA + m∠ABD = 180
m∠3 + 21 + (85 + 17) = 180
m∠3 = 57

Wasn't that a (triangular) piece of cake?