### Classifying Triangles and Quadrilaterals Worksheets

A triangle is formed by joining three non-collinear points using a segment. These segments are called sides and the common endpoints are called vertices. These vertices are used to name or denote a triangle. The triangle below can be named triangle ABC, $$\triangle CBA$$ and $$\triangle BAC$$ with angles: $$\angle A$$, $$\angle B$$, $$\angle C$$and sides: $$\overline{AB}$$, $$\overline {AC}$$ and $$\overline {BC}$$. Sides of triangles can also be named using small letters like a, b, or c.

A triangle can further be classified according to sides and angles.

According to Sides

1. Scalene triangle – a triangle with no two sides equal. The three sides have different lengths. Triangle ABC below is a scalene triangle.
2. Isosceles triangle – a triangle with at least two sides congruent. Triangle ABC below is an isosceles triangle.
3. Equilateral triangle – a triangle with all sides congruent. Triangle GHI is equilateral.
According to Angles

1. Acute triangle – a triangle whose angles are all acute. Triangle ABC below is acute.
2. Obtuse triangle – a triangle with one obtuse angle. Triangle LMN below is obtuse.

3. Right triangle – a triangle with one right angle. Triangle FGH below is right.

4. Equiangular triangle – a triangle with all its angles are congruent. Triangle ABC below is equiangular.
Triangles can even further be classified according to both its angles and sides. The diagram below shows all these possible triangle combinations.

Quadrilaterals are polygons with four sides, four angles and four vertices. In the quadrilateral below, the sides are $$\overline{AB}$$, $$\overline{BC}$$, $$\overline{CD}$$, $$\overline{DA}$$; the angles are $$\angle A$$, $$\angle B$$, $$\angle C$$, $$\angle D$$ and similar to triangles, the vertices of a quadrilateral are used to name it. The quadrilateral below can be named as quadrilateral ABCD, BCDA, CDAB and DABC.
Quadrilaterals can be classified by the special characteristics and relationships between its angles and sides. Listed below are the common types of quadrilateral:

1. Parallelogram

A quadrilateral is a parallelogram if and only if both pairs of its opposite sides area parallel. Several properties of parallelogram were also discovered. These include the following:
1. Opposite sides are congruent
2. Opposite angles are congruent
3. Consecutive angles are supplementary
4. Diagonal separates it into two congruent triangles
5. Diagonals bisect each other
2. Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of sides parallel.

3. Rectangle

A rectangle is a special parallelogram with four right angles. Being a parallelogram, it also has the above mentioned properties of parallelograms.

4. Rhombus

A rhombus is a parallelogram with all sides congruent. Again, being a parallelogram, it also has the above mentioned properties of parallelograms.

5. Square

A square is a rectangle with all sides congruent. A square has the properties of both rectangle and rhombus. Thus, a square is also a rhombus.

6. Trapezium

A trapezium, also known as the general quadrilateral is a quadrilateral with no two sides parallel.

7. Kite

A kite is a quadrilateral with two consecutive sides congruent. Both squares and rhombus have two consecutive sides congruent, thus both are as well considered as kites.
A quadrilateral may also be classified as convex or concave. It is concave if it has an angle that measures more than 180o. If otherwise, it is concave. Figure 1 below is convex while Figure 2 is concave.
Practice Exercise

A. Identify the following.
1. a triangle having three equal sides
2. a triangle whose angles are all acute
3. a triangle with one right angle
4. a triangle whose angles are congruent
5. the longest side of a right triangle

B. Determine whether the following statement is true or false.
1. A right triangle can be an isosceles triangle.
2. It is possible for a triangle to have two obtuse angles.
3. If the three angles of a triangle measure 60 each, the triangle is an acute triangle.
4. No obtuse triangle is equiangular.
5. All equilateral triangles can be a right triangle.

C. Identify the following.
1. a triangle having three equal sides
2. a triangle whose angles are all acute
3. a triangle with one right angle
4. a triangle whose angles are congruent
5. the longest side of a right triangle

D. Determine whether the following statement is true or false.
1. A right triangle can be an isosceles triangle.
2. It is possible for a triangle to have two obtuse angles.
3. If the three angles of a triangle measure 60 each, the triangle is an acute triangle.
4. No obtuse triangle is equiangular.
5. All equilateral triangles can be a right triangle.

E. Name the following.
1. In quadrilateral ABCD, $$\overline {AB}$$ and $$\overline {CD}$$ are __________.
2. In trapezoid DEFG, $$\overline {DE}//\overline{FG}$$ then $$\overline {DE}$$ and $$\overline {FG}$$ are the ________.
3. EFGH is a trapezoid where $$\overline {EF}//\overline{GH}$$ and $$\overline {FG}//\cong \overline{EH}$$ , then EFGH is a/an______.
4. WXYZ is a parallelogram where $$\overline {WX}\cong \overline{XY}$$ and $$\angle XYZ$$ is a right angle, then WXYZ is a______.
5. A parallelogram having one right angle.
6. A figure whose sum of interior angles equals $$360^o$$.
7. A quadrilateral that has one pair of sides parallel.
8. MNOP is a parallelogram. If $$\overline {NO}\cong \overline{OP}$$, then MNOP is a ______.
10. The parallel sides of a trapezoid

F. Determine whether the following statement is true or false.
1. Every parallelogram is a rhombus.
2. Every square is a rectangle.
3. Every rectangle is a parallelogram.
4. Every trapezoid is a parallelogram.
5. Every trapezoid is a quadrilateral.
6. A square is both a rectangle and a rhombus.
7. A rectangle is a quadrilateral and a parallelogram.
8. Every square is a rhombus.
9. Every rhombus is a parallelogram.
10. Every quadrilateral is a trapezium.