Training Lesson: Adjacent and Vertical Angles

MBOU "Secondary School No. 19 with Advanced Study of Certain Subjects"

Training Lesson
"Adjacent and Vertical Angles"

2012-2013 School Year

Training Lesson
"Adjacent and Vertical Angles"

Lesson objectives: to reinforce students' knowledge and skills in applying the definitions and properties of adjacent and vertical angles, develop attention and memory, skills to analyze, compare, and generalize; foster interest in geometry.
Equipment: posters "Adjacent Angles", "Vertical Angles", task cards, notebooks with printed base, proverbs "Wisdom without a guess is worth nothing", "The more I know, the more I can".

Lesson procedure:

Organizational moment: So, we are again in the world of Geometry. In a world where we learn to think, reason, and logically and consistently analyze, seeking simple and beautiful solutions, training our memory and attention.
The topic of our lesson is "Adjacent and Vertical Angles". Your task is to demonstrate how well you know the definitions and properties of adjacent and vertical angles and how well you can apply them.

Preparation of students for educational and cognitive activity.

Theoretical warm-up:

Students receive tasks on cards:

• Prove the theorem of adjacent angles.

• Prove the theorem of vertical angles (based on the poster).

• Among the pictures, find the one needed for proving the theorem of vertical angles, and formulate this theorem.

In the world of "Geometry", it is very important to be able to look and see, notice and identify different features of geometric figures. Develop and train your geometric vision!

Questions for the whole class:

• Name the vertical angles.

• Why are they vertical? (The sides of one angle are supplementary rays of the other, by definition.)

• Are the angles in Figure 4 vertical because they are equal? Is this statement correct? (No, because the sides of one angle are not supplementary rays of the other.)

Do you think the angles shown in the pictures are adjacent? (Yes, in Figure 5 these angles are adjacent, because they have one common side, and their other sides are supplementary rays.)

Thus, by using the definition, we determined whether the angles are adjacent or vertical, i.e. the definition contains characteristics of angles, and their properties are found in the theorems. Now, let’s hear what properties angles have. The class listens and reviews the proofs of the theorems.

Consolidating knowledge and methods of action:
We solve problems without drawings.
For this, let’s mentally imagine the figure:

• If one of the adjacent angles is obtuse, what is the other angle? (Acute, because the sum of adjacent angles equals 180°).

• One of the two angles formed by two intersecting lines is 60°. What are the other angles? Think! (60°, 60°, 120°, 120°)

• Will the angles be adjacent if one is 20° and the other is 160°, and the common part of the two angles is the side? (Yes)

• Can the sum of three angles formed by two intersecting lines be 100°? (No)

• One of the two angles formed by two intersecting lines is 9 times smaller than the other. Find the angles.

Solution:

Let x be the degree measure of the second angle. Then, 9x is the degree measure of the first angle.
The sum of adjacent angles equals 180°.

x + 9x = 180,
10x = 180, x = 18
Answer: 18°, 162°

The task is solved with commentary and checked using a code projector.

The difference between two angles formed by two intersecting lines is 36°. Prove that they are not vertical.

Proof:

Let’s assume they are vertical.
Then, according to the property of vertical angles, they are equal, i.e., the difference is zero.
This contradicts the condition because the difference is 36°.
Therefore, they are not vertical.

Students solve this task independently and then verify it using a code projector.

Applying knowledge and methods of action:

Independent work:
Students complete the tasks in their notebooks with printed bases (Geometry, Grade 7, Saratov, "Licey" publishing house).

Additional task:

Three lines intersect at point O.
Find the sum of the angles: L1 + L2 + L3

Reserve task:
Given: ∟AOB = 50°
∟MOF = 70°
Find ∟AOC, ∟BOD, ∟MOC, ∟COD.

Lesson summary:

Well done! You worked hard and felt the joy of your efforts.
Grades:
"5" –
"4" –
"3" –
Thank you!

Homework:
Textbook "Geometry, 7th – 9th grades", 2002.
Authors: L.S. Atanasyan et al.
#4(3), 8; #19 – additional task, p. 31
#67, #82a; #83 – additional task, questions 17,18, p. 25-27.

Reflection: Let's evaluate our knowledge.
How did we work during the lesson?
Red circle – "5"
Green circle – "4"
Blue circle – poor.