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Lesson 4: Plane Geometric Figures

A. Estimates of plane angle measures

The symbol ◻ at vertex P means that ∠BPC measures 90°. If we divide the opening of ∠BPC into three equal parts, as shown below, then m∠SPC=30° ; m∠APS=30° ; and m∠BPA=30°.

Ray PM divides ∠BPC into two congruent measures, therefore m∠BPM=45° and m∠MPC=45°.

Let us extend line PC to the left and locate point X on it.

• What is the measure of ∠BPX? Why?
• Can you estimate and illustrate 30°, 45°, 60°, 90° and 120° using parts of your body?

Given: Line PY is inclined at 30° with the horizontal. Line BX is perpendicular to line PY at point A.

(BX ⊥ PY at A)

Complete the statements below:

1.  m∠BAY = _________ . Why?

2.  m∠2 + m∠3 + m∠4 = __________ . Why?

3.  ∠1 ≅ __________ . Why?

B. Polygons

A polygon is a closed plane figure bounded by line segments.

The distance around any polygon is called its perimeter.

Which of the following figures below are not polygons? Why?

A regular polygon is a polygon with congruent sides.

Kinds of polygons according to the number of their sides:

 1.  Triangle, a three-sided polygon

 2.  Quadrilateral, a four-sided polygon

 3.  Pentagon, a five-sided polygon

 4.  Hexagon, a six-sided polygon

 5.  Heptagon, a seven-sided polygon

 6.  Octagon, an eight-sided polygon

 7.  Nonagon, a nine-sided polygon

 8.  Decagon, a ten-sided polygon

 9.  Undecagon, an eleven-sided polygon

10. Dodecagon, a twelve-sided polygon

A. Triangles

• A triangle is formed when three noncollinear points are joined by line segments.
• The sum of the three interior angles of a triangle is 180°.
• The total exterior angles of a triangle and all other polygons is 360°.
• A triangle is named by its three vertices like triangle ABC or ∆ABC.

Kinds of triangles according to their sides;

 1.  Equilateral Triangle

An equilateral triangle is a triangle with three sides congruent.

Equilateral triangles are also equiangular, which means that m∠A = m∠B = m∠C.

 2.  Isosceles Triangle

An isosceles triangle is a triangle with two congruent sides.

Base angles of isosceles triangles are congruent.

 3.  Scalene Triangle

A scalene triangle is a triangle with no sides congruent.

∆RST is a scalene triangle.

Answer the following:

 1. A side of an equilateral triangle is 20 centimeters. What is the length of its perimeter?

 2. The upper vertex angle of an isosceles triangle measures 80°. What is the measure of a base angle?

 3. If ∆PXY is equiangular, what is the measure of ∠P?

Previous Lessons

• An Introduction to Plane Geometry -- Part 1
• An Introduction to Plane Geometry -- Part 2
• An Introduction to Plane Geometry -- Part 3