Cumulative Standards Review
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What can you say about the angles by looking at the graph?
See solution.
These angles are also supplementary. Recall that supplementary angles are two angles whose measures have a sum of 180. Therefore, we can conclude that the measures of ∠ 1 and ∠ 2 add to 180. \begin{gathered} \underline\textbf{Statement}\\ \angle 1 \text{ and } \angle 2 \text{ are supplementary}\\ \text{because it is given.} \\ \text{ By the definition of supplementary angles,}\\ m \angle 1 + m \angle 2 =180. \end{gathered} Since m ∠ 1= m ∠ 2, then we can use the Substitution Property of Equality and substitute m ∠ 2 for m ∠ 1 in m ∠ 1 + m ∠ 2 =180. \begin{gathered} \underline\textbf{Statement}\\ m \angle 1= m \angle 2 \text{ and } m \angle 1 + m \angle 2 =180. \\ \text{Substitute } {\color{#009600}{m \angle 2}} \text{ for } m \angle 1. \\\text{ By the Substitution Property of Equality, }\\ {\color{#009600}{m \angle 2}} + m \angle 2 =180. \end{gathered} Notice that if we simplify the left-hand side we will get 2 m ∠ 2=180. Then, we can use the Division Property of Equality and divide both sides by 2, which will give us m ∠ 2 =90. \begin{gathered} \underline\textbf{Statement}\\ \text{Simplify the left-hand side.} \\ \text{By the Division Property of Equality, } \\ m \angle 2 =90. \end{gathered} Recall that ∠ 1 and ∠ 2 are congruent, thus m ∠ 1 is also 90. Since an angle with a measure of 90 is a right angle, then we can conclude that both ∠ 1 and ∠ 2 are right angles. This is what we wanted to prove! \begin{gathered} \underline\textbf{Statement}\\ \angle 1 \cong \angle 2, \text{ so } m \angle 1 =90. \\ \text{Angles with a measure of }90 \text{ are right angles,} \\ \text{so }m \angle 1 \text{ and } m \angle 2 \text{ are right angles.} \end{gathered}
Given:& ∠ 1 and ∠ 2 are supplementary. & ∠ 1 and ∠ 2 are vertical. Prove:& ∠ 1 and ∠ 2 are right angles Proof: ∠ 1 and ∠ 2 are vertical because it is given. By the Vertical Angles Theorem, they are congruent. Since ∠ 1 ≅ ∠ 2, then m ∠ 1=m∠ 2. ∠ 1 and ∠ 2 are supplementary because it is given. By the definition of supplementary angles, we know the following. m ∠ 1+ m∠ 2 =180 m ∠ 1 =m ∠ 2 and m ∠ 1+ m ∠ 2 =180. Substitute m ∠ 2 for m ∠ 1. By the Substitution Property of Equality, we know the following. m ∠ 2+ m∠ 2 =180 Simplify the left-hand side. By the Division Property of Equality, m ∠ 2=90. ∠ 1 ≅ ∠ 2, so m ∠ 1=90. Angles with a measure of 90 are right angles, so m ∠ 1 and m ∠ 2 are right angles.