Question

Journal; 1.8.4, geometry

2. Looking at the Consecutive Interior Angles (9 points)

Look at the diagram of the scenario below. A steep downhill ski slope is intersected at an angle by a less steep ski slope. Safety fences need to be set up in the locations shown. The angles of the fences, angles 1 and 2, can be determined by finding the relationship between the angles a and b.


Draw a geometric diagram of this scenario using two parallel lines and one transversal. (Remember that a transversal is a line which cuts across parallel lines.) Label the angles, parallel lines, and transversal as indicated in the diagram above. (2 points)


Starting with the fact that angles 1 and a are a linear pair and that angles b and 2 are also a linear pair, use a two column proof to prove that consecutive interior angles a and b are supplementary. (5 points)
Statement Reason


Explain what the result of your proof tells you about angles a and b. Specifically, if you measured one angle, what would you know about the other? (2 points)


3. The Exterior Angles (6 points)

The fences will be aligned with the exterior angles ∠1 and ∠2. What are some other relationships you can see between ∠1, ∠2, ∠a, and ∠b? (2 points)

Which of the relationships you listed above will be the most helpful in figuring out the measurements of the safety fences? (2 points)

What is the measure of ∠2? (2 points)


4. Reflections (2 points: 1 point each)
Can you think of any other real-life scenarios where parallel lines and transversals exist?



What are the limitations of the ski slope scenario as a real-life example?

Answer 1

The measurement of the angles formed between parallel lines having a common transversal are readily determined by the relationship between the angles

The measures of the angles in the figures are as follows;

2. The required two column proof is presented as follows;

Statement  ${}$                                                  Reason

∠1 and ∠a are linear pair angles   ${}$             Given

∠1 + ∠a = 180° ${}$                                             Linear pair ∠s are supplementary

∠2 and ∠b are linear pair angles   ${}$            Given

∠2 + ∠b = 180°                                             Linear pair ∠s are supplementary

∠a and ∠b are consecutive interior ∠s   ${}$   Definition

x║y    ${}$                                                           Given

∠1 and ∠b are corresponding angles  ${}$      Definition

∠1 ≅ ∠b   ${}$                                                     Corr. ∠s formed between║ lines

∠1 = ∠b   ${}$                                                      Definition of congruency

∠a + ∠b = 180°                                              Substitution property

∠a and ∠b are supplementary  ${}$                  Definition

The information obtained from the proof is that given ∠1 and ∠b are located in corresponding locations relative to the common transversal of the parallel lines, they (∠1 and ∠b) are equal, and therefore, the the sum of ∠a and ∠b is 180°, given that the sum of ∠1 and ∠a is 180° by substitution property of equality. Which gives that ∠a and ∠b are supplementary angles. Therefore

• If the measure of one of the angles ∠a or ∠b is known, the other angle  is given by subtracting the known angle from 180°

3. Some of the other relationships between ∠1, ∠2, ∠a, and ∠b, are;

• ∠1 and ∠b are corresponding angles, therefore, ∠1 ≅ ∠b, by corresponding angles formed between parallel lines
• ∠2 and ∠a are corresponding angles, therefore, ∠2 ≅ ∠a, by corresponding angles formed between parallel lines
• ∠1 and ∠2 are same side exterior angles and they are therefore supplementary. ∠1 + ∠2 = 180°

• The relationship that will be most helpful in figuring the measurement of the safety fences is the relationship between ∠1 and ∠2; ∠1 + ∠2 = 180°

• The measure of ∠2 = 180° - ∠1

4. A real-life scenario where parallel lines and transversal exist are the handrail and baluster on railings of staircases

The limitations of the Ski slope scenario are;

• The direction of the trail of the skier are usually along or parallel to the slope

• The slope is wide and usually lacks obstruction, and therefore, has few transversals

• The ski slope is usually located away from or follows a different direction from other ski slopes

Learn more about parallel lines having a common transversal here:

https://brainly.com/question/2188951

Answer 2

Answer:

The measurement of the angles formed between parallel lines having a common transversal are readily determined by the relationship between the angles

The measures of the angles in the figures are as follows;

2. The required two column proof is presented as follows;

Statement                                                    Reason

∠1 and ∠a are linear pair angles                Given

∠1 + ∠a = 180°                                              Linear pair ∠s are supplementary

∠2 and ∠b are linear pair angles               Given

∠2 + ∠b = 180°                                             Linear pair ∠s are supplementary

∠a and ∠b are consecutive interior ∠s      Definition

x║y                                                               Given

∠1 and ∠b are corresponding angles        Definition

∠1 ≅ ∠b                                                        Corr. ∠s formed between║ lines

∠1 = ∠b                                                         Definition of congruency

∠a + ∠b = 180°                                              Substitution property

∠a and ∠b are supplementary                    Definition

The information obtained from the proof is that given ∠1 and ∠b are located in corresponding locations relative to the common transversal of the parallel lines, they (∠1 and ∠b) are equal, and therefore, the the sum of ∠a and ∠b is 180°, given that the sum of ∠1 and ∠a is 180° by substitution property of equality. Which gives that ∠a and ∠b are supplementary angles. Therefore

If the measure of one of the angles ∠a or ∠b is known, the other angle  is given by subtracting the known angle from 180°

3. Some of the other relationships between ∠1, ∠2, ∠a, and ∠b, are;

∠1 and ∠b are corresponding angles, therefore, ∠1 ≅ ∠b, by corresponding angles formed between parallel lines

∠2 and ∠a are corresponding angles, therefore, ∠2 ≅ ∠a, by corresponding angles formed between parallel lines

∠1 and ∠2 are same side exterior angles and they are therefore supplementary. ∠1 + ∠2 = 180°

The relationship that will be most helpful in figuring the measurement of the safety fences is the relationship between ∠1 and ∠2; ∠1 + ∠2 = 180°

The measure of ∠2 = 180° - ∠1

4. A real-life scenario where parallel lines and transversal exist are the handrail and baluster on railings of staircases

The limitations of the Ski slope scenario are;

The direction of the trail of the skier are usually along or parallel to the slope

The slope is wide and usually lacks obstruction, and therefore, has few transversals

The ski slope is usually located away from or follows a different direction from other ski slopes

Step-by-step explanation:

AP3X