Math Problem Statement
What are the missing reasons in the proof for proving ABCD is a parallelogram?
Solution
To solve the given problem and identify the missing reasons in the proof, let's analyze the provided statements and reasons.
Given:
Prove:
- is a parallelogram.
Proof Table:
\hline \text{Statement} & \text{Reason} \\ \hline 1. \ m \angle A = 50^\circ, m \angle B = 130^\circ, m \angle C = 50^\circ, m \angle D = 130^\circ & \text{Given} \\ \hline 2. \ m \angle A + m \angle B + m \angle C + m \angle D = 360^\circ & ? \\ \hline 3. \ 50^\circ + 130^\circ + 50^\circ + 130^\circ = 360^\circ & \text{Substitution} \\ \hline 4. \ m \angle D = 130^\circ & \text{Subtraction property of equality} \\ \hline 5. \ \angle A \text{ is congruent to } \angle C & \text{Angles with equal measure are congruent} \\ \hline 6. \ \angle B \text{ is congruent to } \angle D & \text{Angles with equal measure are congruent} \\ \hline 7. \ ABCD \text{ is a parallelogram} & ? \\ \hline \end{array}$$ **Step-by-Step Analysis:** 1. **Statement 2:** - The reason for this statement is that the sum of the interior angles of a quadrilateral is $$360^\circ$$. - Thus, the missing reason for statement 2 is the "Sum of interior angles of a quadrilateral." 2. **Statement 7:** - The criteria for a quadrilateral to be a parallelogram include that opposite angles are congruent. - Since $$\angle A \cong \angle C$$ and $$\angle B \cong \angle D$$, by the definition and properties of a parallelogram, ABCD is a parallelogram. - Thus, the missing reason for statement 7 is the "Converse of opposite angles theorem." **Conclusion:** - The correct answers are: - For statement 2: Sum of interior angles of a quadrilateral. - For statement 7: Converse of opposite angles theorem. Based on these conclusions, the correct answer is: **D. Sum of interior angles; converse of opposite angles theorem** Do you want details or have any questions? Here are some related questions you might find helpful: 1. How can you prove that a quadrilateral is a parallelogram using its sides? 2. What are the properties of a parallelogram? 3. How do you prove the sum of interior angles in a polygon? 4. What is the converse of the opposite angles theorem? 5. How do you determine the measure of an angle in a parallelogram if one angle is given? **Tip:** In geometry, understanding the properties and theorems related to different shapes can significantly simplify proving complex statements.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Geometry
Parallelogram Properties
Angle Relationships
Formulas
Sum of interior angles of a quadrilateral = 360°
Theorems
Sum of Interior Angles Theorem
Converse of Opposite Angles Theorem
Suitable Grade Level
Grade 9-10
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