Math Problem Statement
Solution
The proof shown in the image aims to demonstrate that lines using properties of angles. Here is a breakdown of the proof's steps based on the image:
Given:
This states that the measures of angles and add up to 180°, suggesting they are supplementary.
Prove:
Proof Outline:
Statements | Reasons |
---|---|
Given | |
Vertical Angles Theorem | |
Definition of Congruence | |
Substitution (from previous steps) | |
Angle 6 and angle 7 are supplementary | Definition of Supplementary |
Consecutive Interior Angles Converse |
Explanation:
-
: This is the given information stating that these two angles are supplementary.
-
: Angles 4 and 6 are congruent by the Vertical Angles Theorem, which states that opposite angles formed by two intersecting lines are congruent.
-
: By the Definition of Congruence, congruent angles have equal measures, so .
-
Substitution: Since , we substitute for in the given equation, resulting in .
-
Supplementary: By the Definition of Supplementary, if two angles add up to 180°, they are supplementary. Thus, and are supplementary.
-
Consecutive Interior Angles Converse: The final step uses the Consecutive Interior Angles Converse theorem. When two lines are cut by a transversal and the consecutive interior angles are supplementary, the lines are parallel. Hence, .
Would you like further details or clarifications? Here are some follow-up questions you might consider:
- How is the Vertical Angles Theorem applied in this proof?
- What is the significance of the Consecutive Interior Angles Converse theorem?
- Could a different pair of angles be used to prove ?
- Why is the Substitution step important in this proof?
- How do you know when to use the Definition of Supplementary?
Tip: Always start a geometry proof by identifying given information and known theorems that can help establish the required result.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Geometry
Angles
Parallel Lines
Formulas
Supplementary Angles: ∠4 + ∠7 = 180°
Theorems
Vertical Angles Theorem
Definition of Congruence
Definition of Supplementary
Consecutive Interior Angles Converse
Suitable Grade Level
Grades 8-10