https://file.aiquickdraw.com/mathbot/file/1727990873053o0yn77q6.jpg

Let's break down this proof step by step. The goal is to prove that \( m \angle 1 = m \angle 3 \). The diagram shows two angles, \( \angle DHF \) and \( \angle GHE \), with the given condition that \( m \angle DHF = m \angle GHE \).

Here is how we can assign the reasons for each step:

1. **Statement**: \( m \angle DHF = m \angle 1 + m \angle 2 \)  
   **Reason**: This is the **Angle Addition Postulate**, which states that if a point lies inside an angle, then the measure of the larger angle is the sum of the two smaller angles inside it. In this case, \( \angle DHF \) is divided into \( \angle 1 \) and \( \angle 2 \).

2. **Statement**: \( m \angle GHE = m \angle 2 + m \angle 3 \)  
   **Reason**: Again, this is the **Angle Addition Postulate**, but for \( \angle GHE \), which is divided into \( \angle 2 \) and \( \angle 3 \).

3. **Statement**: \( m \angle DHF = m \angle GHE \)  
   **Reason**: This is **Given**. It was provided in the problem's conditions.

4. **Statement**: \( m \angle 1 + m \angle 2 = m \angle 2 + m \angle 3 \)  
   **Reason**: This follows from **Substitution**. Since \( m \angle DHF = m \angle GHE \), and by the Angle Addition Postulate, you can substitute in the sums from statements 1 and 2.

5. **Statement**: \( m \angle 1 = m \angle 3 \)  
   **Reason**: This follows from **Subtraction Property of Equality**. Subtract \( m \angle 2 \) from both sides of the equation in step 4 to isolate \( m \angle 1 \) and \( m \angle 3 \).

---

Would you like more details on any of the steps, or do you have questions?

Here are five related questions to deepen your understanding:

1. Why does the Angle Addition Postulate hold for any angle?
2. How does substitution work when dealing with angle measures?
3. What properties of equality are used when manipulating angle measures?
4. Can this proof be extended to prove other relationships in the figure?
5. How would the proof change if the given condition were different?

**Tip**: Always identify given information first in a proof, as it provides the foundation for applying properties like substitution or transitivity.

Complete the proof by choosing the correct 'Reason' for the statements based on the given angles: m∠DHF = m∠GHE, and prove that m∠1 = m∠3.

This problem demonstrates the proof of angle equality using the Angle Addition Postulate, Substitution, and Subtraction Properties of Equality. Learn how to break down geometry proofs step-by-step, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation