6.06 Proving angle relationships
Previously, we applied properties of angle relationships in order to solve for unknown values in diagrams. What we didn't mention was that the angle relationships that we applied rely on our assumption of a certain postulate - a statement that we assume to be true.
Angle addition postulate
It might seem intuitive that the measures of two smaller angles add up to the measure of the larger angle they make up. Formally, this relationship is referred to as the angle addition postulate. It's a relationship similar to the segment addition postulate. However, it applies to angles.
Point $B$B lies in the interior of $\angle AOC$∠AOC if and only if $m\angle AOB+m\angle BOC=m\angle AOC$m∠AOB+m∠BOC=m∠AOC.
$B$B lies in the interior of $\angle AOC$∠AOC
$m\angle AOB+m\angle BOC=m\angle AOC$m∠AOB+m∠BOC=m∠AOC
We can apply the angle addition postulate as well as the properties of equality to prove angle measurements in a diagram.
Practice question
Question 1
Consider the attached diagram. Solve for $x$x, given that $m\angle PQR=145^\circ$m∠PQR=145°. Justify your steps using a two-column proof.
Angle relationship theorems
We can apply the angle addition postulate as well as the definitions learned in angle relationships to prove the following theorems about linear pairs and complementary angles:
If two angles form a linear pair, then they are supplementary.
$\angle ABD$∠ABD and $\angle DBC$∠DBC are supplementary because they form a linear pair.
If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary.
$\angle ABD$∠ABD and $\angle DBC$∠DBC are complementary because they form a right angle.
Recall that supplementary angles have measures adding to $180^\circ$180° and complementary angles have measures adding to $90^\circ$90°.
Practice question
Question 2
Use the attached diagram to prove the linear pair theorem.
We can also apply the properties of congruence to help us prove congruence relationships for complementary and supplementary angles.
Angles supplementary to the same angle or to congruent angles are congruent.
Angles complementary to the same angle or to congruent angles are congruent.
Practice question
Question 3
Prove the following part of the congruent complements theorem:
Angles complementary to the same angle are congruent.
We can use the above theorems to help shorten the number of steps needed to complete proofs of other theorems. For example, we can apply the congruent supplements theorem in our proof of the vertical angles congruence theorem.
If two angles are vertical angles, then they are congruent.
Practice question
Question 4
Without directly using the vertical angles congruence theorem, prove that $\angle1\cong\angle3$∠1≅∠3.
Theorems about right angles
Finally, we can use all of the above theorems, as well as the properties of congruence, to prove the following theorems about right angles.
Perpendicular lines intersect to form four right angles.
All right angles are congruent.
Perpendicular lines form congruent adjacent angles.
If two angles are congruent and supplementary, then each angle is a right angle.
If two congruent angles form a linear pair, then they are right angles.
Later, we'll build upon these theorems to prove even more theorems!