Question

match the following items.
given:
△abc
ac = bc
prove:
∠3 = ∠1

1. ac = bc
2. ∠1 = ∠2
3. ∠2 = ∠3
4. ∠1 = ∠3

base ∠s of isosceles triangle =
vertical angles =
given
substitution

Explanation:

Step1: Match \(AC = BC\)

Since \(AC = BC\) is given in the problem statement, it should be matched with "Given".

Step2: Match \(\angle1=\angle2\)

In \(\triangle ABC\) with \(AC = BC\), by the property that base angles of an isosceles triangle are equal, \(\angle1\) (angle at \(A\)) and \(\angle2\) (angle at \(B\) opposite to \(AC\) and \(BC\) respectively) are base angles. So \(\angle1=\angle2\) is matched with "Base \(\angle\)'s of isosceles triangle =".

Step3: Match \(\angle2=\angle3\)

\(\angle2\) and \(\angle3\) are vertical angles. Vertical angles are equal. So \(\angle2 = \angle3\) is matched with "Vertical angles =".

Step4: Match \(\angle1=\angle3\)

Since \(\angle1=\angle2\) and \(\angle2=\angle3\), by substitution (if \(a = b\) and \(b=c\), then \(a = c\)), \(\angle1=\angle3\) is matched with "Substitution".

Answer:

1. \(AC = BC\) \(

ightarrow\) Given

1. \(\angle1=\angle2\) \(

ightarrow\) Base \(\angle\)'s of isosceles triangle =

1. \(\angle2=\angle3\) \(

ightarrow\) Vertical angles =

1. \(\angle1=\angle3\) \(

ightarrow\) Substitution