The geometry tutorial investigates the definitions and properties of triangles. Topics included are isosceles, equilateral, and right triangles. In addition, the sine and cosine laws of triangles are defined.

In this lesson students apply the properties of equilateral triangles. Students examine guided notes, review guided practice, watch instructional videos and attempt practice problems.

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Learn how to construct regular polygons.

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Use interactive geometry software to construct an equilateral triangle that maintains its shape even if you move its vertices.

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Constructions and Angles

Unit Overview

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Use a protractor and ruler.
Identify different types of triangles and quadrilaterals and their characteristics.

Lesson Flow

After an initial exploratory lesson involving a paper folding activity that gets students thinking in general about angles and figures in a context, the unit is divided into two concept development sections. The first section focuses on types of angles—adjacent, supplementary, complementary, and vertical—and how they are manifested in quadrilaterals. The second section looks at triangles and their properties, including the angle sum, and how this affects other figures.

In the first set of conceptual lessons, students explore different types of angles and where the types of angles appear in quadrilaterals. Students fold paper and observe the angles formed, draw given angles, and explore interactive sketches that test many cases. Students use a protractor and ruler to draw parallelograms with given properties. They explore sketches of parallelograms with specific properties, such as perpendicular diagonals. After concluding the investigation of the angle types, students move on to the next set.

In the second set of conceptual development lessons, students focus on triangles. Students again fold paper to create figures and certain angles, such as complementary angles.

Students draw, using a protractor and ruler, other triangles with given properties. Students then explore triangles with certain known and unknown elements, such as the number of given sides and angles. This process starts with paper folding and drawing and continues with exploration of interactive sketches. Students draw conclusions about which cases allow 0, 1, 2, or an infinite number of triangles. In the course of the exploration, students discover that the sum of the measure of the interior angles of a triangle is 180°. They also learn that the sum of the measures of the interior angles of a quadrilateral is 360°. They explore other polygons to find their angle sum and determine if there is a relationship to angle sum of triangles. The exploration concludes with finding the measure of the interior angles of regular polygons and speculating about how this relates to a circle.

Lastly, students solve equations to find unknown angle measures. Using their previous experience, students find the remaining angle measures in a parallelogram when only one angle measure is given. Students also play a game similar to 20 Questions to identify types of triangles and quadrilaterals. Having completed the remaining lessons, students have a four-day Gallery to explore a variety of problems.

The unit ends with a unit assessment.

Students learn to classify triangles by the size of the angles and the measures of the sides.Key ConceptsTriangles are polygons with three sides.Scalene triangles have all sides with a different length and all angles with a different measure.Isosceles triangles have two sides with the same length and two angles with the same measure.Equilateral triangles have all sides with the same length and all angles with the same measure.Acute triangles have all angles with a measure less than 90°.Obtuse triangles have one angle with a measure greater than 90°.Right triangles have one angle with a measure of 90°.ELL: Keep in mind that consistency at the beginning is very important as students begin to learn and apply math vocabulary.Goals and Learning ObjectivesExplore conditions that result in triangles.Identify types of triangles based on the measure of the angles or the measures of the sides.

In this video tutorial, students will determine if a polygon is regular by examining the sides and angles for congruency. Check understanding by the assessment. [2:06]

This tutorial provides practice in classifying triangles which is useful when figuring out how to solve for various sides and angles. [7:15]

If you know what kind of triangle you're looking at, it's much easier to figure out how to solve for various sides and angles. Use this tutorial to practice classifying triangles. [7:15]