Question

problem solving

1. a group of campers tie up their food between two parallel trees, as

shown. the rope is pulled taut, forming a straight line. find ( m angle 2 ). explain
your reasoning. (see example 5.)

proving a theorem
a2) use the reason bank to prove the consecutive interior angles theorem.
given ( p parallel q ), and ( t ) as the transversal
prove ( angle 1 ) and ( angle 2 ) are supplementary angles.

two-column proof

reasons bank definition of supplementary angles given definition of congruent angles substitution property of equality linear pair postulate alternative interior angles theorem

Explanation:

Problem 17 (Mathematics - Geometry)

Step1: Use the property of parallel lines and transversal

Since the two trees are parallel and the rope is a transversal, we know that the angle adjacent to \(76^{\circ}\) and \(\angle2\) are related. The angle adjacent to \(76^{\circ}\) (let's call it \(\angle a\)) forms a linear - pair with \(76^{\circ}\). So, \(m\angle a=180^{\circ}- 76^{\circ}\) (by the Linear - Pair Postulate).

$$m\angle a = 180^{\circ}-76^{\circ}=104^{\circ}$$

Step2: Use the property of alternate interior angles

The two trees are parallel. The rope (transversal) makes \(\angle2\) and \(\angle a\) alternate interior angles. By the Alternate Interior Angles Theorem, if two parallel lines are cut by a transversal, then alternate interior angles are congruent. So, \(m\angle2=m\angle a\)

1. Statement 1: The reason is "Given" as it is the information provided at the start of the proof.
2. Statement 2: \(\angle1\cong\angle3\) because of the "Alternate Interior Angles Theorem" (when two parallel lines \(p\parallel q\) are cut by a transversal \(t\), alternate interior angles are congruent).
3. Statement 3: \(m\angle1\cong m\angle3\) is by the "Definition of congruent angles" (if two angles are congruent, then their measures are equal).
4. Statement 4: \(m\angle2 + m\angle3=180^{\circ}\) is by the "Linear Pair Postulate" (\(\angle2\) and \(\angle3\) form a linear pair).
5. Statement 5: \(m\angle2 + m\angle1=180^{\circ}\) is by the "Substitution Property of Equality" (since \(m\angle1 = m\angle3\), we substitute \(m\angle3\) with \(m\angle1\) in \(m\angle2 + m\angle3=180^{\circ}\)).
6. Statement 6: \(\angle1\) and \(\angle2\) are supplementary angles is by the "Definition of supplementary angles" (if \(m\angle A+m\angle B = 180^{\circ}\), then \(\angle A\) and \(\angle B\) are supplementary).

Answer:

\(m\angle2 = 104^{\circ}\)

Problem A2 (Mathematics - Geometry)