Question

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given ( a parallel b ), and ( c ) is not parallel to ( a ) or ( b ), which statements must be true?
select each correct answer.
( square mangle7 = mangle11 )
( square mangle5 = mangle12 )
( square mangle1 = mangle5 )
( square mangle4 = mangle5 ).

Explanation:

Step1: Analyze ∠7 and ∠11

∠7 and ∠11 are vertical angles (opposite angles formed by intersecting lines b and c). Vertical angles are equal. So m∠7 = m∠11.

Step2: Analyze ∠5 and ∠12

∠5 and ∠12: ∠5 is on line b, ∠12 is on line c. Since c is not parallel to b, no guarantee of equality. So m∠5 ≠ m∠12 (not necessarily true).

Step3: Analyze ∠1 and ∠5

a || b, and the transversal intersects both. ∠1 and ∠5 are corresponding angles. Corresponding angles of parallel lines are equal. So m∠1 = m∠5.

Step4: Analyze ∠4 and ∠5

∠4 and ∠5: ∠4 is on line a, ∠5 is on line b. They are consecutive interior angles? No, ∠4 and ∠5 are not supplementary or equal (∠4 and ∠3 are supplementary, ∠3 and ∠5 are corresponding if a||b? Wait, ∠4 and ∠5: ∠4 is adjacent to ∠3 (supplementary), ∠3 and ∠5 are corresponding (equal). So ∠4 + ∠5 = 180°, not equal. So m∠4 ≠ m∠5.

Answer:

m∠7 = m∠11, m∠1 = m∠5