Question

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Explanation:

Problem 6

Step1: Identify Segment Addition

For segment \( \overline{EFGH} \), by the Segment Addition Postulate, \( EG = EF + FG \) and \( FH = FG + GH \).

Step2: Substitute Given Equality

Given \( EG = FH \), substitute the expressions from Step 1: \( EF + FG = FG + GH \).

Step3: Subtract \( FG \) from Both Sides

Subtract \( FG \) from both sides of the equation: \( EF + FG - FG = FG + GH - FG \), which simplifies to \( EF = GH \).

Problem 7

Step1: Midpoint Definition

Since \( C \) is the midpoint of \( \overline{BD} \), by the definition of a midpoint, \( BC = CD \).

Step2: Segment Addition Postulate

For segment \( \overline{ABCDE} \), by the Segment Addition Postulate:

• \( AE = AB + BC + CD + DE \)
• \( AC = AB + BC \)
• \( CE = CD + DE \)

Step3: Substitute \( AB = DE \) and \( BC = CD \)

Given \( AB = DE \), substitute \( DE \) with \( AB \) in the expression for \( CE \): \( CE = CD + AB \). But \( BC = CD \), so \( CE = BC + AB \).

Step4: Compare \( AC \) and \( CE \)

From Step 2, \( AC = AB + BC \), and from Step 3, \( CE = AB + BC \). Thus, \( AC = CE \), so \( C \) is the midpoint of \( \overline{AE} \) (by definition of a midpoint).

Problem 8

Step1: Definition of Complementary Angles

By definition, if two angles are complementary, their measures sum to \( 90^\circ \). So:

• \( m\angle 1 + m\angle 2 = 90^\circ \) (since \( \angle 1 \) and \( \angle 2 \) are complementary)
• \( m\angle 3 + m\angle 4 = 90^\circ \) (since \( \angle 3 \) and \( \angle 4 \) are complementary)

Step2: Substitute \( m\angle 2 = m\angle 4 \)

Given \( m\angle 2 = m\angle 4 \), substitute \( m\angle 4 \) with \( m\angle 2 \) in the second equation: \( m\angle 3 + m\angle 2 = 90^\circ \).

Step3: Set Equations Equal

From Step 1, \( m\angle 1 + m\angle 2 = 90^\circ \), and from Step 2, \( m\angle 3 + m\angle 2 = 90^\circ \). Thus, \( m\angle 1 + m\angle 2 = m\angle 3 + m\angle 2 \).

Step4: Subtract \( m\angle 2 \) from Both Sides

Subtract \( m\angle 2 \) from both sides: \( m\angle 1 + m\angle 2 - m\angle 2 = m\angle 3 + m\angle 2 - m\angle 2 \), which simplifies to \( m\angle 1 = m\angle 3 \).

Answer:

(Proof completed as follows)