given: adjacent angles $\angle a$ and $\angle b$, formed by the intersection...
prove: at least one of the angles $\angle a$ and $\angle b$ has measure $90^\circ$ or greater

proof:
first, we assume that this conclusion is false. in other words, we assume that the contrary statement
\select one ? has measure select one ?\
is ?.

this assumption is equivalent to the following two statements:
(1) $m\angle a$ ??$^\circ$ and
(2) $m\angle b$ ??$^\circ$.

using (1) and (2) and addition properties of inequalities, we conclude that $m\angle a + m\angle b$ ? $180^\circ$.

on the other hand, two adjacent angles form a linear pair. thus, the last statement contradicts the select one ? which states that for a linear pair of angles $\angle a$ and $\angle b$, $m\angle a + m\angle b$ ? $180^\circ$.

therefore, the assumption made is ? and the statement

Question

given: adjacent angles $\angle a$ and $\angle b$, formed by the intersection...
prove: at least one of the angles $\angle a$ and $\angle b$ has measure $90^\circ$ or greater

proof:
first, we assume that this conclusion is false. in other words, we assume that the contrary statement
\select one ? has measure select one ?\
is ?.

this assumption is equivalent to the following two statements:
(1) $m\angle a$ ??$^\circ$ and
(2) $m\angle b$ ??$^\circ$.

using (1) and (2) and addition properties of inequalities, we conclude that $m\angle a + m\angle b$ ? $180^\circ$.

on the other hand, two adjacent angles form a linear pair. thus, the last statement contradicts the select one ? which states that for a linear pair of angles $\angle a$ and $\angle b$, $m\angle a + m\angle b$ ? $180^\circ$.

therefore, the assumption made is ? and the statement

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Indirect Proof", "Linear Pair Postulate" ], "new_concepts": [], "current_concepts": [ "Indirect Proof", "Linear Pair Postulate", "Properties of Inequalities" ] } ## Negate the conclusion for indirect proof $$ \begin{aligned} &\text{Conclusion to prove: At least one of the angles } \angle A \text{ and } \angle B \text{ has measure } 90^\circ \text{ or greater.}\ &\text{Contrary assumption: "both angles } \angle A \text{ and } \angle B \text{" has measure "less than } 90^\circ\text{".}\ &\text{Equivalent statements:}\ &(1)\ m\angle A < 90^\circ\ &(2)\ m\angle B < 90^\circ \end{aligned} $$ ## Sum the inequalities and identify the contradiction $$ \begin{aligned} &m\angle A + m\angle B < 180^\circ\ &\text{Linear Pair Postulate states that for a linear pair of angles } \angle A \text{ and } \angle B,\ m\angle A + m\angle B = 180^\circ.\ &\text{The inequality } m\angle A + m\angle B < 180^\circ \text{ contradicts the Linear Pair Postulate.} \end{aligned} $$ ## Conclude the proof $$ \begin{aligned} &\text{Therefore, the assumption made is false, and the statement is true.} \end{aligned} $$ Given: Adjacent angles $\angle A$ and $\angle B$, formed by the intersection... Prove: At least one of the angles $\angle A$ and $\angle B$ has measure $90^\circ$ or greater. **Proof:** First, we assume that this conclusion is false. In other words, we assume that the contrary statement " both angles $\angle A$ and $\angle B$ has measure less than $90^\circ$ " is true. This assumption is equivalent to the following two statements: (1) $m\angle A$ $<$ $90^\circ$ and (2) $m\angle B$ $<$ $90^\circ$. Using (1) and (2) and addition properties of inequalities, we conclude that $m\angle A + m\angle B$ $<$ $180^\circ$. On the other hand, two adjacent angles form a linear pair. Thus, the last statement contradicts the Linear Pair Postulate which states that for a linear pair of angles $\angle A$ and $\angle B$, $m\angle A + m\angle B$ $=$ $180^\circ$. Therefore, the assumption made is false and the statement is true. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Indirect Proof" ] }

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QUESTION IMAGE

Question

Explanation:

Negate the conclusion for indirect proof

Sum the inequalities and identify the contradiction

Conclude the proof

Answer: