Lesson 1.1: Understanding Conditional Statements in Proofs
Learn how to use conditional statements, converses, inverses, and contrapositives in geometric proofs.
Example: If a shape is a square, then it has four equal sides.
Solution:
Contrapositive: If a shape does not have four equal sides, then it is not a square.
What is the inverse of "If a figure is a triangle, then it has three sides."?
A) If a figure is not a triangle, then it has three sides.
B) If a figure is not a triangle, then it does not have three sides.
C) If a figure has three sides, then it is a triangle.
D) If a figure is a triangle, then it does not have three sides.
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Explanation:
The **inverse** of a conditional statement negates both the hypothesis and the conclusion.
Original: If a figure is a triangle, then it has three sides.
Inverse: If a figure is not a triangle, then it does not have three sides.
The correct answer is **B) If a figure is not a triangle, then it does not have three sides.**
Visualizing Conditional Statements
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Lesson 1.2: Writing Two-Column Proofs
Understand how to structure geometric proofs using statements and reasons.
Example: Proving that vertical angles are congruent.
Solution:
Step 1: Given two intersecting lines, identify vertical angles.
What are the two parts of a two-column proof?
A) Equations and numbers
B) Statements and reasons
C) Postulates and theorems
D) Assumptions and guesses
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Explanation:
A **two-column proof** consists of **statements** (steps taken to prove a theorem) and **reasons** (justifications for each step).
Example:
Statements
Reasons
1. ∠A and ∠B are vertical angles
Given
2. Vertical angles are congruent
Vertical Angle Theorem
The correct answer is **B) Statements and reasons.**
Visualizing Vertical Angles
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Lesson 1.3: Triangle Congruence Theorems
Learn the criteria for proving triangles congruent (SSS, SAS, ASA, AAS, HL).
Example: Prove two triangles congruent using SSS.
Solution:
Step 1: Identify three pairs of congruent sides.
Which theorem proves two right triangles congruent with a hypotenuse and one leg?
A) SSS Theorem
B) HL Theorem
C) ASA Theorem
D) AAS Theorem
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Explanation:
The **Hypotenuse-Leg (HL) Theorem** states that two **right triangles** are congruent if their **hypotenuse** and **one leg** are congruent.
This is a special case of the **SSS Theorem** that applies only to right triangles.
The correct answer is **B) HL Theorem.**
Visualizing Triangle Congruence
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