What are the smallest and the biggest angles you can create from adding two acute angles?
An acute, a right or an obtuse angle.
Practice makes perfect
Let's look at the different classifications of angles and see if we can make them out of two smaller angles.
Can we form an acute angle?
We have two acute angles which we can call ∠ A_1 and ∠ A_2. We add them together and form a new angle. Let's call this ∠ S. If m∠ A_1=20^(∘) and m∠ A_2=30^(∘) it looks like this.
Let's now add these two angles together to form ∠ S. The Angle Addition Postulate tells us that m∠ S=50^(∘) which is an acute angle.
Can we form a right angle?
Let's now look at the special case where m∠ A_1=20^(∘) and m∠ A_2=70^(∘).
We will now add these two angles together. Again, we use the Angle Addition Postulate to calculate m∠ S=90^(∘) which is a right angle.
Hence, we can add two acute angles and form a right angle.
Can we form an obtuse angle?
Let's continue with letting m∠ A_1=20 ^(∘) and m∠ A_2=80^(∘).
We now continue to form ∠ S. The Angle Addition Postulate tells us that m∠ S = 100^(∘) which is obtuse.
Can we form a straight angle?
Let's now investigate if it is possible to add two acute angles and form a straight angle. Let's do this by starting out with ∠ S and let it have a measure of 180^(∘).
If we bisect ∠ S we find two angles, ∠ A_1 and ∠ A_2, we can add together to form ∠ S. If we bisect an angle the resulting angles will have the same measure, m, which is half that of the original angle.
m=180^(∘)/2=90^(∘)
The resulting angles will therefore both be right.
We now know that two right angles will form a straight angle.
Conclusion
If one of the angles is acute it is smaller than a right angle. The another angle then would have to be larger than 90^(∘), which is an obtuse angle. It is therefore not possible to add two acute angles to form a straight angle. Let's now summarize. If we add to acute angles we can form the following.
