First write a proof that there can be at most one right angle in a triangle. Then, write a proof that there can be at most one obtuse angle in a triangle.
See solution.
Practice makes perfect
We want to prove that there can be at most one right or obtuse angle in a triangle. Let's write a proof that there can be at most one right angle in a triangle first. Then, we will write a proof that there can be at most one obtuse angle in a triangle.
One Right Angle
Let's consider a triangle that has one right angle, ∠ C.
We can begin our proof by stating the given information.
Statement
In △ ABC, ∠ C is a right angle.
By the definition of a right angle, we can conclude that the measure of ∠ C is 90.
Statement
By the definition of a right angle, m ∠ C=90.
We can use the Triangle Angle-Sum Theorem and write that the sum of the measures of the angles in △ ABC is 180.
Statement
By the Triangle Angle-Sum Theorem, m ∠ A + m ∠ B + m ∠ C = 180.
Now, we can substitute 90 for m ∠ C in our equation.
Statement
By the Substitution Property of Equality,
m ∠ A+ m ∠ B +90 = 180.
By the Subtraction Property of Equality, we can subtract 90 from both sides of the equation.
Statement
By the Subtraction Property of Equality,
m ∠ A+ m ∠ B =90.
Finally, notice that if the measure of one of the angles A or B was 90, then the other one would be 0. This is impossible, so there cannot be two right angles in a triangle.
Statement
If m ∠ A=90, then m ∠ B=0.
If m ∠ B=90, then m ∠ A=0.
It is impossible that the measure of one of the angles in a triangle is0.
Therefore, there is at most one right angle in a triangle.
Completed Proof
There can be at most one right angle in a triangle.
Proof: In △ ABC, ∠ C is a right angle. By the definition of a right angle, m ∠ C =90. By the Triangle Angle-Sum Theorem,
m ∠ A+ m ∠ B + m ∠ C = 180.
By the Substitution Property of Equality, m ∠ A + m ∠ B + 90 = 180. By the Subtraction Property of Equality, m ∠ A + m ∠ B = 90. If m ∠ A = 90, then m ∠ B=0. If m ∠ B = 90, then m ∠ A = 0. It is impossible that the measure of one of the angles in a triangle is 0. Therefore, there is at most one right angle in a triangle.
One Obtuse Angle
Let's consider a triangle that has one obtuse angle, ∠ C.
We can begin our proof by stating the given information.
Statement
In △ ABC, ∠ C is an obtuse angle.
By the definition of an obtuse angle, we can conclude that the measure of ∠ C is greater than 90.
Statement
By the definition of an obtuse angle, m ∠ C >90.
We can use the Triangle Angle-Sum Theorem and write that the sum of the measures of the angles in △ ABC is 180.
Statement
By the Triangle Angle-Sum Theorem, m ∠ A + m ∠ B + m ∠ C = 180.
Since m ∠ C>90, we can conclude that m ∠ A + m ∠ B <90. Therefore, ∠ A and ∠ B must both be acute.
Statement
We know that m ∠ C >90, so m ∠ A + m ∠ B < 90.
Therefore, ∠ A and ∠ B are acute angles.
Completed Proof
There can be at most one obtuse angle in a triangle.
Proof: In △ ABC, ∠ C is an obtuse angle. By the definition of an obtuse angle, m ∠ C >90. By the Triangle Angle-Sum Theorem, m ∠ A + m ∠ B + m ∠ C = 180. We know that m ∠ C >90, so m ∠ A+ m ∠ B <90. Therefore, ∠ A and ∠ B are acute angles.
