Topicm4c684b47f2476970_1528449000663_0Topic

Angles and diagonals in quadrilaterals

Levelm4c684b47f2476970_1528449084556_0Level

Third

Core curriculumm4c684b47f2476970_1528449076687_0Core curriculum

VIII. Plane geometry. The student:

4) applies the properties of angles and diagonals in rectangles, parallelograms, rhombuses and trapezoids.

Timingm4c684b47f2476970_1528449068082_0Timing

45 minutes

General objectivem4c684b47f2476970_1528449523725_0General objective

Noticing regularities, similarities and analogies and formulating relevant conclusions.

Specific objectivesm4c684b47f2476970_1528449552113_0Specific objectives

1. Applying the triangle angle sum theorem.

2. Determining geometric relationships in quadrilaterals.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm4c684b47f2476970_1528450430307_0Learning outcomes

The student:

- proves and uses the triangle angle sum theorem,

- determines geometric relationships in quadrilaterals.

Methodsm4c684b47f2476970_1528449534267_0Methods

1. Observation.

2. Discussion.

Forms of workm4c684b47f2476970_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm4c684b47f2476970_1528450127855_0Introduction

The teacher gives each pair of students a worksheet with the following figure.

[Illustration 1]

Students formulate and prove the triangle angle sum theorem based on this figure.

Theorem – the triangle angle sum theorem.

- The sum of the angles in a triangle is equal to 180°.

Students discuss the ways of applying this theorem for determine geometric relationships in quadrilateral.

Procedurem4c684b47f2476970_1528446435040_0Procedure

The teacher divides students into 5 groups. The task of each group is to formulate the properties of angles and diagonals in:

1) a squaresquaresquare,
2) a rectanglerectanglerectangle,
3) a parallelogramparallelogramparallelogram,
4) a rhombusrhombusrhombus,
5) a trapezoidtrapezoidtrapezoid.

Auxiliary questions for the given quadrilateral:

1. Is the sum of the angles on one side of the quadrilateral constant? How large is it?
2. Are the opposite angles equal?
3. At what angle do the diagonals intersect?
4. Are the diagonals of equal length?
5. Are the diagonals bisectors of angles?m4c684b47f2476970_1527752263647_01. Is the sum of the angles on one side of the quadrilateral constant? How large is it?
2. Are the opposite angles equal?
3. At what angle do the diagonals intersect?
4. Are the diagonals of equal length?
5. Are the diagonals bisectors of angles?

Task
Students can use the Geogebra applet - Angles and diagonals in quadrilaterals.

[Geogebra applet]

Group work should be summarized by formulating the following properties:

1. In the parallelogramparallelogramparallelogram, the sum of consecutive anglesconsecutive anglesconsecutive angles is equal to 180°.
2. In the parallelogram, the opposite anglesopposite anglesopposite angles are congruent.
3. The diagonals of a parallelogram bisect each other.
4. The diagonals in a rhombusrhombusrhombus are the bisectors of an angle.
5. In a trapezoidtrapezoidtrapezoid the sum of angles at the leg is equal to 180°.

Students work individually and then discuss the results.

Task
In the parallelogram ABCD the diagonal AC is the bisector of the angle BAD.
Show that diagonals AC and BD in this parallelogram intersect at the right angle.m4c684b47f2476970_1527752256679_0In the parallelogram ABCD the diagonal AC is the bisector of the angle BAD.
Show that diagonals AC and BD in this parallelogram intersect at the right angle.

Task
1. The shorter diagonal divides the right trapezoid into two right isosceles triangles. Find the measure of the angles of this trapezoidtrapezoidtrapezoid.
2. The side and the shorter diagonaldiagonaldiagonal in the rhombusrhombusrhombus have the same length a. Find the length of the longer diagonal.

An extra task:
The diagonals in a parallelogramparallelogramparallelogram intersect at the 60° angle and one of them is two times shorter than the other. Find the measure of the angle between the shorter side of the parallelogram and its longer diagonaldiagonaldiagonal.

Lesson summarym4c684b47f2476970_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they summarize the class by formulating the main relationships to memorize.

1. In the parallelogramparallelogramparallelogram the sum of consecutive anglesconsecutive anglesconsecutive angles is equal to 180°.
2. In the parallelogram the opposite anglesopposite anglesopposite angles are congruent.
3. The diagonals of a parallelogram bisect each other (they are mutually bisecting).
4. The diagonals in the rhombusrhombusrhombus are the bisectors of its angle.
5. In the trapezoidtrapezoidtrapezoid the sum of the angles at the leg is equal to 180°.

Selected words and expressions used in the lesson plan

consecutive anglesconsecutive anglesconsecutive angles

diagonaldiagonaldiagonal

diagonals bisect each otherdiagonals bisect each otherdiagonals bisect each other

opposite anglesopposite anglesopposite angles

parallelogramparallelogramparallelogram

rectanglerectanglerectangle

rhombusrhombusrhombus

squaresquaresquare

trapezoidtrapezoidtrapezoid