Question

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question #1
based on the figure shown, sandra concluded that ∠1 ≅ ∠7.
which relationship correctly justifies sandra’s conclusion?
a ( mangle1 + mangle4 = 180^circ ) and ( angle4 cong angle7 )
b ( mangle1 + mangle2 = 180^circ ) and ( angle2 cong angle7 )
c ( angle1 cong angle8 ) and ( angle8 cong angle7 )
d ( angle1 cong angle5 ) and ( angle5 cong angle7 )
(figure with intersecting lines and labeled angles 1,2,3,4,5,6,7,8)

Explanation:

Step1: Analyze Vertical Angles/Transversal

From the figure, two lines intersect (vertical angles) and a transversal crosses them. Let's check each option.

Step2: Evaluate Option D

Option D: \( \angle 1 \cong \angle 5 \) (corresponding angles, since lines are parallel? Wait, or vertical/transversal. Wait, \( \angle 5 \cong \angle 7 \) (vertical angles). So if \( \angle 1 \cong \angle 5 \) and \( \angle 5 \cong \angle 7 \), then by transitive property, \( \angle 1 \cong \angle 7 \). Let's check other options:

• Option A: \( m\angle 1 + m\angle 4 = 180^\circ \) (supplementary, linear pair), but \( \angle 4 \cong \angle 7 \)? No, \( \angle 4 \) and \( \angle 7 \) – not necessarily.
• Option B: \( m\angle 1 + m\angle 2 = 180^\circ \) (linear pair), \( \angle 2 \cong \angle 7 \)? No.
• Option C: \( \angle 1 \cong \angle 8 \) (no, \( \angle 1 \) and \( \angle 8 \) – not corresponding. \( \angle 8 \cong \angle 7 \)? No, \( \angle 8 \) and \( \angle 6 \) are vertical, \( \angle 7 \) and \( \angle 5 \) vertical. Wait, maybe I misread. Wait, the correct reasoning: \( \angle 1 \) and \( \angle 5 \) are corresponding angles (if lines are parallel, but even as vertical/transversal, \( \angle 5 \) and \( \angle 7 \) are vertical angles (so \( \angle 5 \cong \angle 7 \)). So \( \angle 1 \cong \angle 5 \) (corresponding) and \( \angle 5 \cong \angle 7 \) (vertical), so \( \angle 1 \cong \angle 7 \) (transitive). So Option D is correct.

Answer:

D. \( \angle 1 \cong \angle 5 \) and \( \angle 5 \cong \angle 7 \)