Topicmf00019464cb64ca1_1528449000663_0Topic

Supplementary anglessupplementary anglesSupplementary angles, vertical anglesvertical anglesvertical angles

Levelmf00019464cb64ca1_1528449084556_0Level

Third

Core curriculummf00019464cb64ca1_1528449076687_0Core curriculum

VIII. Plane geometry. The student:

12) carries out geometric proofs.

Timingmf00019464cb64ca1_1528449068082_0Timing

45 minutes

General objectivemf00019464cb64ca1_1528449523725_0General objective

Reasoning, including multiple‑stage arguments, giving arguments, justifying the correctness of reasoning, distinguishing a proof from an example.

Specific objectivesmf00019464cb64ca1_1528449552113_0Specific objectives

1. Application of properties of supplementary anglessupplementary anglessupplementary angles for solving problems, including problems to prove theorems.

2. Application of vertical anglesvertical anglesvertical angles properties for solving problems, including problems in order to prove a theorem.

Learning outcomesmf00019464cb64ca1_1528450430307_0Learning outcomes

The student:

- uses properties of supplementary anglessupplementary anglessupplementary angles for solving problems, including problems to prove theorems,

- application of vertical anglesvertical anglesvertical angles properties for solving problems, including problems to prove theorems.

Methodsmf00019464cb64ca1_1528449534267_0Methods

1. Discussion.

Forms of workmf00019464cb64ca1_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmf00019464cb64ca1_1528450127855_0Introduction

The teacher informs students that in the lesson they will use the properties of supplementary and vertical anglesvertical anglesvertical angles to solve geometric problems, including problems to prove a theorem.

Students review their knowledge of definition and properties of supplementary and vertical anglesvertical anglesvertical angles.

The supplementary anglessupplementary anglessupplementary angles are the two angles that have one ray, and the other rays are complementary to the straight line.

The vertical anglesvertical anglesvertical angles are two angles that have a common vertex and the rays of one angle are the corresponding rays of the other angle.

The sum of measures of supplementary anglessupplementary anglessupplementary angles is equal to 180°.

The measures of vertical angle are equal.

Note: In English, two terms are used, which are similarly translated into Polish. Supplementary anglessupplementary anglesSupplementary angles are adjacent anglesadjacent anglesadjacent angles. Adjacent angles are adjacent (neighbouring) angles, and this term is not used in Polish mathematical terminology. The adjacent corners are angles with one and only one common ray. The sum of supplementary  angles is 180°, the sum of measures of supplementary anglessupplementary anglessupplementary angles is equal to the angle determined by the remaining rays. Thus, the supplementary anglessupplementary anglessupplementary angles are a special case of adjacent anglesadjacent anglesadjacent angles.

Proceduremf00019464cb64ca1_1528446435040_0Procedure

Students work in pairs, using computers, practise the ability to calculate angles measures, using the properties of vertical and supplementary anglessupplementary anglessupplementary angles.

Task
Open the Geogebry applet - Vertical anglesvertical anglesVertical angles and supplementary anglessupplementary anglessupplementary angles. Do the tasks. Repeat them several times to solve them flawlessly.

[Geogebra applet]

Students work in groups, carrying out geometric proofs using the properties of supplementary and vertical anglesvertical anglesvertical angles.

Task - Group 1
Using only the properties of supplementary anglessupplementary anglessupplementary angles, show that the measures of the vertical anglesvertical anglesvertical angles are equal.

Task - Group 2
Remembering that the triangle angle measure is equal to 180°, show that for the angles shown in the figure, the equality is $\gamma =\alpha +\beta$.

[Illustration 1]

Task - Group 3
Proof that the angles shown in the picture meet the following condition $\delta =\beta +\gamma .$

[Illustration 2]

Task - Group 4
Using the picture below proof that, if $\alpha =\beta$ thus $\gamma +\delta ={180}^{\circ }.$

[Illustration 3]

The groups present prepared proofs of theorems. The teacher draws students’ attention to the correctness of the proof and the correctness of the proof notation.

An extra task:
Prove that the angles shown in the figure meet the condition $\delta =\alpha +\beta +\gamma$.

Tip: 
Extend the section of CD to be intersected with the side AB and mark the point of the intersection as E.

[Illustration 4]

Lesson summarymf00019464cb64ca1_1528450119332_0Lesson summary

Students do the revision exercises.

Then together summarize the class, by formulating the conclusions to memorize.

- The sum of measures of supplementary anglessupplementary anglessupplementary angles is equal to 180°.

- The measures of vertical anglesvertical anglesvertical angles are equal.

Selected words and expressions used in the lesson plan

adjacent anglesadjacent anglesadjacent angles

are equal in measureare equal in measureare equal in measure

measure of anglemeasure of anglemeasure of angle

sum of anglessum of anglessum of angles

supplementary anglessupplementary anglessupplementary angles

vertical anglesvertical anglesvertical angles