Question

given:
\triangle abr
\overrightarrow{rs}\parallel\overrightarrow{ab}
\angle1 = \angle2
prove:
ra = rb
helpful hint: identify the given statement first.

1. line rs \parallel segment ab, \angle1 = \angle2
2. \angle b = \angle1
3. \angle a = \angle2
4. \angle a = \angle b
5. ra = rb

Explanation:

Step1: Identify the reason for ∠B = ∠1

Since \( \overrightarrow{RS}\parallel\overrightarrow{AB}\), by the property of parallel lines, if lines are parallel, corresponding angles are equal. So, \( \angle B=\angle1\) (If lines \( \parallel\), corresponding \( \angle\)'s are \( =\)).

Step2: Identify the reason for ∠A = ∠2

Since \( \overrightarrow{RS}\parallel\overrightarrow{AB}\), by the property of parallel lines, if lines are parallel, alternate - interior angles are equal. So, \( \angle A = \angle2\) (If lines \( \parallel\), alternate interior \( \angle\)'s \( =\)).

Step3: Use substitution

Given \( \angle1=\angle2\), and from step1 \( \angle B=\angle1\), from step2 \( \angle A=\angle2\). By substitution, \( \angle A=\angle B\) (Substitution).

Step4: Apply the isosceles triangle theorem

In \( \triangle ABR\), if two angles (\( \angle A\) and \( \angle B\)) of a triangle are equal, then the sides opposite to them are equal. So, \( RA = RB\) (If two \( \angle\)'s of a triangle are \( =\), sides opposite are \( =\)).

Answer:

1. Given
2. If lines \( \parallel\), corresponding \( \angle\)'s are \( =\)
3. If lines \( \parallel\), alternate interior \( \angle\)'s \( =\)
4. Substitution
5. If two \( \angle\)'s of a triangle are \( =\), sides opposite are \( =\)