prove: $\triangle abc$ has at least one angle with measure less than $45^\circ$

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

we were given that $\angle a$ is obtuse. that is,
(1) $m\angle a$ ? $^\circ$.

furthermore, the contrary statement given above implies
(2) $m\angle b$ ? $^\circ$ and
(3) $m\angle c$ ? $^\circ$.

using (1)-(3) and addition properties of inequalities, we conclude that $m\angle a + m\angle b + m\angle c$ ? $180^\circ$.

but this contradicts the select one ?, which states that $m\angle a + m\angle b + m\angle c$ ? $180^\circ$.

Question

prove: $\triangle abc$ has at least one angle with measure less than $45^\circ$

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

we were given that $\angle a$ is obtuse. that is,
(1) $m\angle a$ ? $^\circ$.

furthermore, the contrary statement given above implies
(2) $m\angle b$ ? $^\circ$ and
(3) $m\angle c$ ? $^\circ$.

using (1)-(3) and addition properties of inequalities, we conclude that $m\angle a + m\angle b + m\angle c$ ? $180^\circ$.

but this contradicts the select one ?, which states that $m\angle a + m\angle b + m\angle c$ ? $180^\circ$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Proof by Contradiction", "Triangle Inequality", "Triangle Angle Sum" ], "current_concepts": [ "Proof by Contradiction", "Triangle Angle Sum Theorem", "Inequalities in Geometry" ] } ## Formulate the contrary statement $$ \begin{aligned} &\text{Original statement: } \triangle ABC \text{ has at least one angle with measure less than } 45^\circ.\ &\text{Contrary statement: } \triangle ABC \text{ has } \text{no angles} \text{ with measure } \text{less than } 45^\circ \text{ is true.}\ &\text{Equivalently: } \triangle ABC \text{ has } \text{all angles} \text{ with measure } \text{greater than or equal to } 45^\circ \text{ is true.} \end{aligned} $$ ## Translate given conditions and contrary statement into inequalities $$ \begin{aligned} &\text{(1) } \angle A \text{ is obtuse } \implies m\angle A > 90^\circ.\ &\text{(2) Contrary statement implies } m\angle B \ge 45^\circ.\ &\text{(3) Contrary statement implies } m\angle C \ge 45^\circ. \end{aligned} $$ ## Sum the inequalities and identify the contradiction $$ \begin{aligned} &m\angle A + m\angle B + m\angle C > 90^\circ + 45^\circ + 45^\circ = 180^\circ.\ &\text{This contradicts the Triangle Angle Sum Theorem, which states that } m\angle A + m\angle B + m\angle C = 180^\circ. \end{aligned} $$ Prove: $\triangle ABC$ has at least one angle with measure less than $45^\circ$ **Proof:** First we assume that this conclusion is false. In other words, we assume that the contrary statement "$\triangle ABC$ has no angles with measure less than $45^\circ$" is true. We were given that $\angle A$ is obtuse. That is, (1) $m\angle A$ $>$ $90$$^\circ$. Furthermore, the contrary statement given above implies (2) $m\angle B$ $\ge$ $45$$^\circ$ and (3) $m\angle C$ $\ge$ $45$$^\circ$. Using (1)-(3) and addition properties of inequalities, we conclude that $m\angle A + m\angle B + m\angle C$ $>$ $180^\circ$. But this contradicts the Triangle Angle Sum Theorem, which states that $m\angle A + m\angle B + m\angle C$ $=$ $180^\circ$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Proof by Contradiction" ] }

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Question

Explanation:

Formulate the contrary statement

Translate given conditions and contrary statement into inequalities

Sum the inequalities and identify the contradiction

Answer: