Geometry / Unit 2: Reasoning and Proof
2.6 Proving Angle Relationships
Notes
Some basic angle facts are proved as theorems and then used as reasons.
- Vertical Angles Theorem: vertical angles are congruent.
- Linear Pair Postulate: if two angles form a linear pair, they are supplementary.
- Congruent Supplements Theorem: angles supplementary to the same angle (or congruent angles) are congruent. The same holds for complements.
- All right angles are congruent.
- Example: $\angle 1$ and $\angle 2$ form a linear pair, and $\angle 2$ and $\angle 3$ form a linear pair. Then $\angle 1 \cong \angle 3$ by Congruent Supplements.
Practice
$\angle A$ and $\angle B$ are both supplementary to $\angle C$. $m\angle A = 65^\circ$. What is $m\angle B$?
Angles supplementary to the same angle are congruent.
Which reason proves that vertical angles $\angle 1$ and $\angle 3$ are congruent when each forms a linear pair with $\angle 2$?
Both are supplementary to $\angle 2$, so they are congruent.
$\angle P$ is a right angle and $\angle Q$ is a right angle. Which conclusion is justified?
All right angles are congruent. Nothing says where they are, so they need not be adjacent or vertical.
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